I expect the most relevant action to most people's lives will be some nasty budget showdowns (likely not affecting ICE, since they're funded for years already and attempting to touch that will not go well) and the really interesting thing to see will be whether Trump's administration attempts to reverse the executive-created (and rather arbitrary and silly) modern idea of what a "shutdown" is, or will attempt to significantly adjust the (again, kinda arbitrary anyway) traditionally categories for what does and doesn't continue to be funded.

AFAIK it's never been argued in court, it's just what someone in the executive branch in the late '70s decided should happen, and before that in the event of failure to pass a budget it was assumed that spending should continue as-is, until another budget is passed, more-or-less. It's not even clear it'd be illegal if he just went on like nothing had happened.