Both happen, my guess is that it is more common to have a written standard that just doesn't match some dialects. Sometimes you will have multiple standards, which is usually due to politics/history.

An example where the same words are written differently is Serbian/Croatian/Bosnian/Montenegrin. One dimension among several that dialects there vary along is the development of a vowel from a common ancestor Slavic language into /e/, /i/, or /(i)je/. This leads to a variety of spellings for some words such as "vreme/vrime/vrijme" (time). It's made even more complicated by the variation crossing borders and even not being consistent within one people. For example standard Serbian has /e/ and the /(i)je/ since Bosnian Serbs mostly use the latter, Bosnian and Croatian use /(i)je/, and /i/ is technically not in any written standard but still exists and can be found in colloquial writing.

Personally if I were to design a writing system from scratch I don't think going "full regular" is all that it's sometimes made out to be. You can lose clues about common roots between words. That said keeping historical spellings forever can also be confusing. Russian Cyrillic has had a number of reforms to trim things that had become redundant/not pronounced anymore. English could use some of that as well, but the extreme variety and lack of a central writing system authority makes that hard.