"be doing a case" sounds OK. "be working a case" sounds weird The best one for me is "be working ON a case" be working on + specific task e.g. "I've been working hard on my English skills recently."

Actually it’s pretty common to drop the preposition after working. He’s working (on) the case. He’s working (at) the front door. Let’s work (on) the problem. There could also be a difference in meaning. For example, He’s working on his biceps. (He’s doing things to strengthen his biceps in general) He’s working his biceps. (Right now, he’s using them) I’m not sure whether, in the case of the police, there is a difference in meaning between ‘working on’ and ‘working’ but they both sound natural to me. But as others have said ‘working on’ is the default. For example I’m working on my homework. (Correct) I’m working my homework. (Incorrect)

Here is a different example: "She works two jobs just to earn enough to survive." In this case, "job" is a role, a position, not just one task or case. Think about "work a job" as a special phrase. Usually, "work" as a verb is followed by a preposition, like in, on (see previous comment) or for

Hi 👋🏼 I hope this information helps you! The sentence "The detective is working ON a murder case" is a natural way to express that the detective is investigating a murder case. However, "The detective is doing a murder" is not a natural way to express that the detective is investigating a murder case. The word "doing" suggests an action that is being performed, rather than an investigation that is being carried out. Additionally, the phrase "a murder" on its own is unclear and potentially misleading as it can refer to the act of committing a murder rather than investigating one. A clearer way to express the same idea would be to say "The detective is investigating a murder." Or “The detective is doing a murder investigation.” Note: “conduct an investigation” is a collocation in English. A collocation is basically a naturally occurring combination of words that are always used side by side in a particular context which means that collocations are by default idiomatic. Examples: ➡️The police are conducting an investigation into the robbery. ➡️ The detective will conduct an investigation to find out who committed the crime.

Although grammatical, it is not common usage to say "work a case" or "do a case". You "work on a case". Likewise, a student works "on his homework". So, the words "case" and "homework" are used in similar ways. Without "on", "work" means you are impacting something. For example, when you make bread by hand, the first thing you have to do is "work the dough". "Do a case" does not work because a case is not any sort of procedure that can be done. A doctor can "do an operation" because operations are procedures. A dancer can "do the cha-cha" because there is a method to the cha-cha that you must learn how to do.