Removing flush mounted ceiling fixture that has springs atached to it
Trim rings for can fixtures (which are recessed, but look "flush" from below) are often spring-mounted. The ones I've met either have a pair of "long-legged" springs that go into a slot where they will release if you squeeze the legs together (the trim pulls down from the ceiling several inches and stops on bent-out ends of the springs) or two or three coil springs, where you usually have to unhook one end of the coil spring to release the trim (which is more of a pain.) 

Either one can be awkward, so start by being on a secure, comfortable work platform, not precariously balanced on whatever is handy.