If you're talking about police/detectives, then 'work a case' is fine to say. In the US, this is a common way to express 'to work on a case.' 'Sargeant, I want to work a case.' 'It's been a while since he's worked a case.'

Hello. My name is Colin, I am an American, English is my first language and I consider my English to be better than that of most Americans. Many Americans will say "irregardless" in a sentence, but "irregardless" is not a word. Many Americans will say "I literally died," ignoring the fact that it's impossible for dead people to speak. They misuse the word "literally." Most Americans overuse adverbs like these: basically, seriously, totally, completely, absolutely, definitely, literally, essentially, entirely I could type more, but my main point is that just because an American uses English a certain way does not mean that that American should be imitated. I've only heard "I'm working on a case." "I'm working a case" sounds to me simply like lazy English.

Most of the answers you received were oversimplified. That’s the instinct of most English teachers. There’s a presumption that learners shouldn’t be bothered with the complexities of English and that ‘correct’ = ‘simple’.

Yes, it's a very interesting problem and allows the introduction of the handy English word 'pedantic.' Especially in regard to colloquial usage, or in this case, what is basically 'trade language' or some kind of form of in-group speech. If cops use the term and it's generally a police term that a learner may encounter repeatedly in TV and film, insisting that the usage is technically incorrect is of limited use and may be confusing. Further, I would say, 'working a case' has extra meaning as it may give extra insight into the mental and physical processes that police undergo when they are 'working a case' rather than the more sterile and possibly less committed sounding 'working on a case.'