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When learning chengyu, at least in Chinese one has to confront the question: do we go with the classical ones taken from the Grand Masters or with nice stories behind them or do we also include Chengyu that are frequently used nowadays but might not hail from a classical source? I guess you just have to bite the bullet and know at least a couple of hundred chengyu, because you need to be able to both demonstrate your command of the classics, but also you have to know the run-of-the-mill chengyu nobody would call especially classic, but which get used all the time nonetheless.

What I've tried is to find a list of 30-40 chengyu that can be considered to be essential, both from the classicist's point of view and the perspective of popular usage. For this, I used the appendix provided by the Chengyu dictionary from the Taiwanese Ministry of Education (MOE), which contains 48,000+ chengyu that are assigned numbers according to how many sources include them (by sources they mean chengyu dictionaries and textbooks and the like, they list them here) . I think that’s a good compromise. The numbers will still be skewed towards ‘classical’ chengyu because that’s what gets put in most compilations and so on, but frequently used chengyu not from classic sources still have a chance (yes even words like 马马虎虎). So I have taken the 37 chengyu from the MOE Chengyu list that appear in 25 or more sources (27 is the highest score in the sample of 48,000+ chengyu) and put them into a list. (the translations are mostly from CEDICT, I looked up the forms that weren’t in CEDICT). I hope though that these 37 chengyu are a good compromise for a learner of Modern Mandarin: frequently used but still considered classical by many.

1. 一丘之貉 yīqiū zhī háo jackals of the same tribe (idiom); fig. They are all just as bad as each other.
2. 掩耳盗铃 yăn'ĕr-dàolíng to plug one's ears while stealing a bell (idiom); to deceive oneself / to bury one's head in the sand
3. 朝三暮四 zhāosān-mùsì lit. say three in the morning but four in the evening (idiom); to change sth that is already settled upon / indecisive / to blow hot and cold
4. 杞人忧天 Qĭrén-yōutiān man of Qǐ fears the sky falling (idiom); groundless fears
5. 一毛不拔 yīmáo-bùbá (saying) stingy; parsimonious
6. 班门弄斧 bānmén-nòngfŭ display one's slight skill before an expert
7. 囫囵吞枣 húlún-tūnzăo to read hastily without thinking
8. 竭泽而渔 jiézé-éryú kill the goose that lays the golden eggs
9. 天衣无缝 tiānyī-wúfèng perfect (job); without a trace; flawloess (lies)
10. 名落孙山 míngluò-sūnshān (idiom) to fail in imperial exam
11. 指鹿为马 zhĭlù-wéimă making a deer out to be a horse (idiom); deliberate misrepresentation
12. 水落石出 shuĭluò-shíchū As the water recedes, the rocks appear (idiom); the truth comes to light / When facts are known, doubts dissipate. / Murder will out. / Name of a CCTV soap opera set in police station
13. 胸有成竹 xiōngyŏu-chéngzhú to plan in advance (idiom); a card up one's sleeve / forewarned is forearmed
14. 举一反三 jŭyī-fănsān to raise one and infer three (idiom); to deduce many things from one case
15. 青出于蓝 qīngchū-yúlán lit. green is born of blue, but beats blue (idiom); fig. the student becomes superior to the master / same as couplet 青出于蓝,而胜于蓝|青出于蓝,而胜于蓝
16. 江郎才尽 Jiāngláng-cáijìn Mr Jiang has exhausted his talent (idiom); to have used up one's literary talent or energy
17. 一败涂地 yībài-túdì failed and wiped over the floor (idiom); to fail utterly / a crushing defeat / beaten and in a hopeless position
18. 信口雌黄 xìnkŏu-cíhuáng to talk off the cuff (idiom); idle talk
19. 口若悬河 kŏuruò-xuánhé mouth like a torrent (idiom) / eloquent / glib / voluble / have the gift of the gab
20. 如火如荼 rúhuŏ-rútú lit. white cogon flower like fire (idiom); fig. a mighty army like wildfire / daunting and vigorous (momentum) / flourishing / magnificent
21. 沧海桑田 cānghăi-sāngtián lit. blue seas where once was mulberry fields (idiom); time brings great changes / life's vicissitudes
22. 为虎作伥 wéihŭ-zuòchāng to act as accomplice to the tiger (idiom); to take the part of the devil / to help a villain against honest people
23. 狐假虎威 hújiă-hŭwēi lit. the fox exploits the tiger's might (idiom); fig. to use powerful connections to intimidate people
24. 狼狈为奸 lángbèi-wéijiān villains collude together (idiom); to work hand in glove with sb (to nefarious ends)
25. 画蛇添足 huàshé-tiānzú lit. draw legs on a snake (idiom); fig. to ruin the effect by adding sth superfluous / to overdo it
26. 老马识途 lăomă-shítú an old horse knows the way (idiom); an experience worker know what to do / an old hand knows the ropes
27. 草木皆兵 căomù-jiēbīng lit. every tree or bush an enemy soldier (idiom); fig. to panic and treat everyone as an enemy / to feel beleaguered
28. 惊弓之鸟 jīnggōng zhī niăo lit. bird startled by a bowshot (idiom); fig. a frightened person
29. 南辕北辙 nányuán-bĕizhé (fig.) act in a way that defeats one's purpose
30. 望梅止渴 wàngméi-zhĭkĕ quench one's thirst by thinking of plums (idiom); to console onself with false hopes
31. 势如破竹 shìrú-pòzhú sweeping
32. 司空见惯 sīkōng-jiànguàn a common occurrence
33. 含沙射影 hánshā-shèyĭng insinuate
34. 投笔从戎 tóubĭ-cóngróng (of student or intellectual) give up civilian pursuits to join the military, to join the army voluntarily
35. 东窗事发 dōngchuāng-shìfā to come to light; the game is up
36. 夜郎自大 Yèláng-zìdà ignorant and boastful
37. 作法自毙 zuòfă-zìbì a scheme that boomerangs; to get into trouble through one own's scheme

These are HSK-listed chengyu:

实事求是 shi2 shi4 qiu2 shi4 be practical and realistic HSK B
力所能及 li4 suo3 neng2 ji2 in one's power HSK B
不好意思 bu4 hao3yi4si shy/ feel embarrassed HSK B
千方百计 qian1 fang1 bai3 ji4 by every possible means/ by hook or crook HSK C
粗心大意 cu1xin1 da4yi4 careless/ inadvertent/ negligent HSK C
各式各样 ge4 shi4 ge4 yang4 of all styles and types/ all kinds of HSK C
一路平安 yi1 lu4 ping2 an1 one-mouth-breath/ in one breath/ at one go HSK C
万古长青 wan4 gu3 chang2 qing1 thousands-age-ever-green/ remain fresh forever HSK C
成千上万 cheng2 qian1 shang4 wan4 thousands of/ large quantity of HSK C
兴高采烈 xing4 gao1 cai3 lie4 spirits-high-mood-ardent/ be in high spirits/ be raptured HSK C
一路顺风 yi1 lu4 shun4 feng1 Safe journey HSK C
多劳多得 duo1 lao2 duo1 de2 more pay for more work HSK C
四面八方 si4 mian4 ba1 fang1 four sides and eight directions/ in every direction HSK C
或多或少 huo4 duo1 huo4 shao3 more or less HSK C
画蛇添足 hua4 she2 tian1 zu2 ruin by adding superfluous HSK C
无论如何 wu2lun4 ru2he2 no-matter-like-what/ in any case/ in any event/ at any rate HSK C
无可奈何 wu2 ke3 nai4he2 nothing-could-make-what/ be helpless HSK C
聚精会神 ju4 jing1 hui4 shen2 with complete concentration/ with rapt attention HSK C
自言自语 zi4 yan2 zi4 yu3 self-talk-self-say/ talk to oneself HSK C
有两下子 you3 liang3 xia4zi have-two-feats/ know a few tricks of the trade HSK C
自始至终 zi1 shi3 zhi4 zhong1 from-beginning-to-end/ from start to finish HSK C
自相矛盾 zi4 xiang1 mao2dun4 self-mutually-spear-shield/ self-contradiction HSK C
这样一来 zhe4yang4 yi1 lai2 this-way-ever-since/ in this way/ in such a case/ thus HSK C
闹着玩儿 nao4zhe wan2r frolic/ be just kidding/ tease/ act in a frivolous manner HSK C
总而言之 zong3 er2 yan2 zhi1 generalize-and-say-it/ in short/ in brief/ in a nutshell HSK C
不怎么样 bu4 zen3meyang4 not so good/ not particularly good/ pretty bad HSK C
走后门儿 zou3 hou4men2r walk-back-door/ gain advantages through connection HSK C
能歌善舞 neng2 ge1 shan4 wu3 be good at both singing and dancing HSK C
轰轰烈烈 hong1hong1 lie4lie4 dynamic/ vigorous HSK D
理所当然 li3 suo3 dang1 ran2 go without saying/ of course HSK D
合情合理 he2 qing2 he2 li3 fair and reasonable HSK D
理直气壮 li3 zhi2 qi4 zhuang4 one is assured and bold with justice HSK D
和平共处 he2ping2 gong4 chu3 peaceful coexistence HSK D
坚贞不屈 jian1zhen1 bu4 qu1 remain faithful and unyielding/ stand firm and unbending HSK D
接二连三 jie1 er4 lian2 san1 one after another/ in quick succession HSK D
浩浩荡荡 hao4hao4 dang4dang4 go forward with great strength and vigor HSK D
津津有味 jin1jin1 you3 wei4 with relish/ with gusto HSK D
精打细算 jing1 da3 xi4 suan4 pinch pennies HSK D
开天辟地 kai1 tian1 pi4 di4 since the beginning of the history/ the creation of the world HSK D
全力以赴 quan2 li4 yi3 fu4 do one's level best/ go all out HSK D
精益求精 jing1 yi4 qiu2 jing1 keep improving HSK D
兢兢业业 jing1jing1 ye4ye4 cautious and conscientious HSK D
敬而远之 jing4 er2 yuan3 zhi1 keep someone at a respectful distance HSK D
举世闻名 ju3shi4 wen2ming2 of world known/ world-famous HSK D
举世瞩目 ju3shi4 zhu3mu4 attract worldwide attention HSK D
门当户对 men2 dang1 hu4 dui4 be well-matched in social and economic status HSK D
美中不足 mei3 zhong1 bu4 zu2 a blemish in an otherwise perfect thing HSK D
可歌可泣 ke3 ge1 ke3 qi4 move one to praises and tears HSK D
可想而知 ke3 xiang3 er2zhi1 you can imagine HSK D
没吃没穿 mei2 chi1 mei2 chuan1 have no food and clothes HSK D
来回来去 lai2 hui2 lai2 qu4 come-and-go HSK D
全心全意 quan2 xin1 quan2 yi4 with all one's heart HSK D
不相上下 bu4 xiang1 shang4 xia4 about the same/ equally matched HSK D
各行各业 ge4 hang2 ge4 ye4 all trades and professions HSK D
从容不迫 cong2rong2 bu4 po4 take one's time/ calm and unhurried HSK D
川流不息 chuan1 liu2 bu4 xi1 flowing up in an endless stream HSK D
诚心诚意 cheng2 xin1 cheng2 yi4 sincere desire HSK D
层出不穷 ceng2 chu1 bu4 qiong2 emerge in endlessly HSK D
不知不觉 bu4 zhi1 bu4 jue2 unconsciously/ unknowingly HSK D
大公无私 da4 gong1 wu2 si1 selfless/ unselfish HSK D
不言而喻 bu4 yan2 er2 yu4 tell its own tale HSK D
大同小异 da4 tong2 xiao3 yi4 largely identical but with minor difference HSK D
不辞而别 bu4 ci2 er2 bie2 leave without saying goodbye HSK D
不卑不亢 bu4bei1bu4kang4 neither haughty not humble HSK D
暴风骤雨 bao4 feng1 zhou4 yu3 violent storm tempest HSK D
半真半假 ban4 zhen1 ban4 jia3 half-genuine; half-sham HSK D
半途而废 ban4 tu2 er2 fei4 give up halfway/ leave sth unfinished HSK D
百家争鸣 bai3 jia1 zheng1 ming2 a hundred schools of thought contend HSK D
不正之风 bu4 zheng4 zhi1 feng1 unhealthy tendency HSK D
发奋图强 fa1fen4 tu2 qiang2 work with a will to make the country strong HSK D
顾全大局 gu4quan2 da4ju2 take the whole account into consideration HSK D
供不应求 gong1 bu4 ying4 qiu2 supply falls short of demand HSK D
根深蒂固 gen1 shen1 di4 gu4 being deeply rooted/ in grain HSK D
各奔前程 ge4 ben4 qian2 cheng2 each goes his own way HSK D
七嘴八舌 qi1 zui3 ba1 she2 all talking at once HSK D
格格不入 ge2 ge2 bu4 ru4 incompatible with/ out of tune with HSK D
大包大揽 da4bao1 da4lan3 undertake the whole thing/ take on every thing HSK D
发扬光大 fa1yang2 guang1da4 carry forward HSK D
归根到底 gui1 gen1 dao4 di3 at all time and place HSK D
家喻户晓 jia1 yu4 hu4 xiao3 widely known HSK D
断断续续 duan4duan4 xu4xu4 off and on/ intermittently HSK D
独立自主 du2li4 zi4zhu3 paddle one's own canoe HSK D
东奔西走 dong1 ben1 xi1 zou3 run around here and there/ rush around HSK D
得不偿失 de2 bu4 chang2 shi1 be not worth the candle/ get more kicks than halfpence HSK D
大有可为 da4 you3 ke3 wei2 be well worth doing/ have bright prospects HSK D
改邪归正 gai3 xie2 gui1 zheng4 give up evil and return to good HSK D
循序渐进 xun2 xu4 jian4 jin4 follow in order and advance step by step HSK D
弄虚作假 nong4 xu1 zuo4 jia3 practice fraud/ resort to deception HSK D
一毛不拔 yi4 mao2 bu4 ba2 miserly/ very stingy/ unwilling to give up even a hair HSK D
一技之长 yi1 ji4 zhi1 chang2 professional skill/ specially HSK D
一哄而散 yi1 hong4 er2 san4 break up in a hubbub HSK D
一个劲儿 yi1gejin4'er persistently HSK D
一干二净 yi1 gan1 er4 jing4 completely/ thoroughly HSK D
饮水思源 yin3 shui3 si1 yuan2 never forget where one's happiness comes from HSK D
一帆风顺 yi1 fan1 feng1 shun4 plain sailing HSK D
引人注目 yin3 ren2 zhu4 mu4 conspicuous/ noticeable HSK D
新陈代谢 xin1 chen2 dai4xie4 metabolism HSK D
欣欣向荣 xin1xin1 xiang4 rong2 flourishing/ prosperous HSK D
小心翼翼 xiao3xin1 yi4yi4 with great care HSK D
想方设法 xiang3 fang1 she4 fa3 do everything possible/ try every means HSK D
显而易见 xian3 er2 yi4 jian4 obviously/ stick out a mile HSK D
无微不至 wu2 wei1 bu4 zhi4 meticulously/ in every possible way HSK D
一概而论 yi1gai4 er2 lun4 lump together al and sundry/ lump under one head HSK D
斩钉截铁 zhan3 ding1 jie2 tie3 resolute and decisive HSK D
自私自利 zi4 si1 zi4 li4 self-concern/ selfish HSK D
自力更生 zi4 li4 geng1 sheng1 self-dependence HSK D
自负盈亏 zi4 fu4 ying2 kui1 assume sole responsibility for its profits or losses HSK D
诸如此类 zhu1 ru2 ci3 lei4 and all that/ and so on HSK D
众所周知 zhong4 suo3 zhou1 zhi1 as everyone knows/ it is common knowledge that HSK D
争先恐后 zheng1 xian1 kong3 hou4 strive to be the first/ vie with each other HSK D
以身作则 yi3 shen1 zuo4 ze2 set oneself an example to others HSK D
朝气蓬勃 zhao1 qi4 peng2 bo2 full of youthful spirit HSK D
无能为力 wu2 neng2 wei2 li4 incapable of action/ powerless HSK D
斩草除根 zhan3 cao3 chu2 gen1 cut the weeds and dig up the roots HSK D
指手画脚 zhi3 shou3 hua4 jiao3 make indiscreet remarks or criticism HSK D
有声有色 you3 sheng1 you3 se4 vivid and dramatic HSK D
有口无心 you3 kou3 wu2 xin1 be sharp-tongued but not malicious HSK D
由此可见 you2 ci3 ke3 jian4 thus it can be seen HSK D
永垂不朽 yong3 chui2 bu4 xiu3 immortal HSK D
朝三暮四 zhao1 san1 mu4 si4 blow hot and cold/ play fast and loose HSK D
萍水相逢 ping2 shui3 xiang1 feng2 meet by chance HSK D
热泪盈眶 re4 lei4 ying2 kuang4 one's eyes brimming with tears HSK D
前所未有 qian2 suo3 wei4 you3 unprecedented HSK D
前赴后继 qian2 fu4 hou4 ji4 advance wave upon wave HSK D
千军万马 qian1 jun1 wan4 ma3 a large number of mounted and foot soldiers HSK D
恰如其分 qia4 ru2 qi2 fen4 just right/ appropriate HSK D
恰到好处 qia4 dao4 hao3 chu4 just right HSK D
无所作为 wu2 suo3 zuo4 wei2 attempt and accomplish nothing HSK D
奇花异草 qi2 hua1 yi4 cao3 exotic flowers and rare herbs HSK D
十全十美 shi2 quan2 shi2 mei3 be perfect in every way/ leave nothing to be desired HSK D
百花齐放 bai3 hua1 qi2 fang4 let a hundred flowers blossom HSK D
目中无人 mu4 zhong1 wu2 ren2 consider everyone beneath one's notice HSK D
莫名其妙 mo4 ming2 qi2 miao4 without rhyme or reason HSK D
名副其实 ming2 fu4 qi2 shi2 be worthy of the name/ the name matches the reality HSK D
面面俱到 mian4 mian4 ju4 dao4 reach every aspect of a matter HSK D
乱七八糟 luan4qi1ba1zao1 in a mess/ at sixes and sevens HSK D
岂有此理 qi3 you3 ci3 li3 outrageous/ preposterous HSK D
讨价还价 tao3 jia4 huan2 jia4 bargain HSK D
连滚带爬 lian2 gun3 dai4 pa2 roll and crawl/ flee in panic HSK D
无可奉告 wu2 ke3 feng4 gao4 no comment HSK D
无话可说 wu2 hua4 ke3 shuo1 have nothing to say HSK D
微不足道 wei1 bu4 zu2 dao4 not worth mentioning/ inconsiderable HSK D
万水千山 wan4 shui3 qian1 shan1 the trials of a long journey HSK D
推来推去 tui1 lai2 tui1 qu4 push/ decline HSK D
如醉如痴 ru2 zui4 ru2 chi1 enjoy (the concert; etc) very much HSK D
天长地久 tian1 chang2 di4 jiu3 everlasting HSK D
三番五次 san1 fan1 wu3 ci4 again and again/ time and again HSK D
滔滔不绝 tao1tao1 bu4 jue2 dash along/ give it mouth HSK D
探头探脑 tan4 tou2 tan4 nao3 pop one's head in and look about HSK D
损人利己 sun3 ren2 li4 ji3 harm others to benefit oneself HSK D
似笑非笑 si4 xiao4 fei1 xiao4 a faint on one's face/ with a spurious smile HSK D
似是而非 si4 shi4 er2 fei1 specious/ apparently right but actually wrong HSK D
思前想后 si1 qian2 xiang3 hou4 think over again and again HSK D
无情无义 wu2qing2wu2yi4 heartless HSK D
投机倒把 tou2ji1 dao3ba3 engage in speculation and profiteering HSK D