As the title says — is "could've" or "should've" standard English or is it slang and should correctly be spelled "could have" and "should have"?
Contractions are generally considered informal, but have long been part of standard English. Because they are informal, most style guides—which tend to be guides for formal styles of written English—advise against using them. This of course makes sense, because if you are trying to write in a formal style, using informal words, like contractions, makes your writing seem less formal.
The Corpus of Historical American English has incidences of should’ve dating to 1910, could’ve dating to 1880, and would’ve dating to 1830. A Google Book Search shows incidences of all three from 1800–1810.
For the record, slang is “very informal usage in vocabulary and idiom that is characteristically more metaphorical, playful, elliptical, vivid, and ephemeral than ordinary language.” Contractions with have are informal, sure, but they are also ordinary, and so are not really slang.
Slang eventually becomes standardized, right?
Anyway, it's not incorrect to write them with the contractions. Writing could've and should've is standard, even if the spellcheckers say differently. The contractions just emphasize the pronunciation of the words.