Looking for very common dialect words that have no Standard Mandarin Chinese hanzi characters
It is probably not the languages/dialects that don't have a corresponding Chinese character, but rather regional slang. The A菜 you see is actually 萵仔菜, or ue-á-tshài in Hokkien. That became became e-á-tshài which led it to be transcribed back into Chinese as A仔菜 and eventually A菜.

There is actually a word for Q, but I am not aware of how to type that out on a computer. So have a picture of it instead:  (EDIT: This has apparently made it into Unicode as U+296A8, "")

Owing to the dominance of Standard Chinese, characters not in common use in that language tend not to make it into a common computer encoding. So in a sense, the dialectal words indeed do not have a corresponding standard Chinese character. On the other hand, that's more of an issue with the way Chinese works on computers.

In any case, this phenomenon tends to happen in cases where Sinitic languages/dialect retain words from Old Chinese rather than the more fashionable new words that have taken over in Standard Mandarin. A couple more examples from Hokkien:


 gâu, meaning "good at, skilled at, prone to"
 lān, meaning "male genitalia"
 in, meaning "them, her/his"
 tsi̍p, meaning "a taste, to treat to a casual meal"
 peh, meaning "to climb"


There are likely hundreds if not thousands of such characters in common use. There might not be a list per se, but such characters can be found in dictionaries that caters to a specific language/dialect.

EDIT: I have a massive blindspot for Taiwanese loanwords.

Due to being under Japanese rule for five decades at a time of rapid change, Taiwanese had a rather unique opportunity at absorbing Japanese terms. Now, I don't know if we can say such words have no Chinese characters per se. Some of them came with kanji characters already. And naturally, they all have Mandarin equivalents, including some which are transcribed into Chinese from the same ultimate (Western) source. Examples:


Aspirin: a33 sir55 phi55 lin51 - Chinese 阿斯匹林, Japanese アスピリン (asupirin)
Clear-cut: at3 sa55 lih3 - Chinese 阿莎力, Japanese あっさり (assari)
Miso: mi55 sooh3 - Chinese 味噌, Japanese 味噌 (miso)
Tempura: thian35 pu55 lah3 - Chinese 甜不辣, Japanese 天麩羅 (tempura)


With Aspirin, both Chinese and Japanese (and hence Taiwanese) loaned the same word by transcribing its sound. The word for "clear-cut", on the other hand, it is a Japanese word that has been adopted in Chinese by transcribing its pronunciation - same as for Taiwanese.

Tempura and Miso are both Japanese words that had kanji. But whereas Chinese adopted Miso, it transcribed Tempura in other Chinese characters. And in both cases, Taiwanese follows the Japanese pronunciation.

Whether or not one considers them to be Taiwanese words without Chinese characters might be a matter of personal opinion. Compare and contrast with the following Taiwanese terms, which don't have such relationships to their equivalents in Chinese characters:


Apron: e33 pu55 long51 - Chinese 圍裙, from エプロン
Butter: ba55 tah3 - Chinese 奶油, from バタ－
Bra: bu33 la51 jia11 - Chinese 胸罩, from ブラジャ－
Condom: sak5 khuh3 - Chinese 保險套 from サック
Gasolin: 29 ga33 soo55 lin51 - Chinese 汽油, from ガソリン
Ice Cream: ai55 sirh3 khu33 lin51 mu11 - Chinese 冰淇淋, from アイスクリ－ム
Bread: phang51 - Chinese 麵包, from パン
Stainless Steel: sir33 ten55 le51 su11 - Chinese 不鏽鋼, from ステンレス


It's possible that this is mirrored in some other dialects/languages. Hong Kong Cantonese comes to mind.

EDIT2:

After some more research I've learnt of a few Taiwanese words where the proper Chinese character was unknown or disputed or perhaps non-existent. This, combined with long term marginalisation of Taiwanese, resulted in some chaos in terms of how some words were written.

As I suspected in the comments, however any words that didn't have a character have been matched with one nonetheless. And in 2009 the government made an effort to standardise the writing system. In some cases, supposedly new characters were created; but so far I haven't found truly new ones.

In any case, here are some words that apparently had no universally agreed upon correct character (prior to 2009, at least:


bē/buē: officially 袂, was/is written as , 未
gâu: officially , was/is written as 賢, 爻
buaih: officially 勿愛 squished together, was/is written as 無愛, 覅
suí: officially 媠, was/is written as 美, 水
tsia̍h: officially 食, was/is written as 噍, 吃
phah: officially 拍, was/is written as 打, 扑
beh/bueh: officially 欲, was/is written as 要, 卜
kah: officially 佮, was/is written as 及, 甲, 合
m̄: officially 毋, was/is written as 不, 呣, 唔
tshù: officially 厝, was/is written as 茨、戍