woodwind it's more than possible - i've seen pieces with quartersharps & flats, quartersharp & flat trills etc. it's a modern thing but it has been (and is being) used alot, and not just for things that can do it easily...
obviously bowed strings can do it by putting your finger somewhere in between your standard equal tempered notes; trombone, half a slide position; voice, just sing when you're really tired

; fretted strings, bend the string; woodwinds & brass, lip up or down, easier with a reed (listen to how jazz musicians bend notes all over the place) but possible on everything; timpani, just turn the tuning things/mess around with the pedal (i dunno).
piano, however, would be tricky... unless you opened the piano up, and, playing with one hand, twist the tuning things on the inside with the other, and you'd get some microtones... although no doubt some ultramodern piano maker has made a piano with a microtone pedal that makes all the notes sharper or flatter (like harp pedals)...