Prompted to post this after hearing the news on RNZ speculation "if Trump fires Mueller".
Trump can't directly fire Mueller, that would be up to the Attorney General, but Sessions has recused himself from the enquiry (and Trump was not happy about that). Deputy AG Rosenstein has the authority, then if he recused himself it steps down again. It's complicated:
There are so many fish hooks Trump could skewer himself on and so much risk, especially as he's now got a twitter-trail where he's almost self-incriminated by comments gloating about (partisan) reasons for vile actions he's had carried out. (ie gloating about firing McCabe, when that was supposed to have been up to the justice department - not an executive decision by POTUS etc) Those fall short (only just) of obstruction of justice - and he could easily tip the balance.
If he goes ahead pressuring the Justice Department to fire Mueller and/or end the enquiry, then I think it's a safe bet that he knows that Mueller has something on him so damning that it will end his presidency. I think he might be cornered / running out of options - but continuing to attack Mueller and the enquiry is probably the safest bet perhaps for now. Long may that continue. I'd rather see Trump lose big time on the mid-terms and stumble the end of his term, hurting the GOP as much as possible with every tweet and rant, than to see Pence take over with his extremely dangerous form of "christian" social conservatism.