For example, if I want to say
他写de是"..."
which "de" should I use?
是 isn't a noun, so that suggests using 得 but on the other hand, I've seen usages of "写的是什么" (on Google search at least)
There are many similar questions on this site.
I think we use 的 in this case.
Because
他写的是...
is a simplified form of
他写的内容是...
内容 is a noun, therefore we can use 的 here.
Your understanding of what differentiates the various de characters is a bit flawed. A simple rule goes like this:
1) If it comes between an adverb and a verb, use 地:
大聲地喊 shout loudly
快地跑 run quickly
2) If it comes between a verb and a complement phrase, use 得:
玩得開心 play until happy
說得不對 say [something] untrue
忙得不得了 busy to an unbearable degree
This sentence construct doesn't have a direct equivalent in English, so you generally need to add words in the translation.
3) In most other situations, use 的:
我的自行車 my bicycle
你喜歡的 that which you like
他想說的是什麼 what does he want to say
So in your sentence, 他寫的 is the subject, meaning "that which he wrote", and 是什麼 is the predicate, meaning "is what?". It can be understood as an abbreviation of 他寫的字.
However, 他寫得是 is also correct, but it means something very different. Here, 是 has the meaning of "correct" and is used as a complement, so this phrase means "He wrote [something] correct".
