What do the Q look like? Do the Q actually have any sort of physical form that humans can perceive, or are they incorporeal?
The Q are(most likely) beyond human perception
In the Voyager episode "Death Wish" the crew is actually brought to the continuum.
The Continuum is manifested in this form so that it will "Fall within [human] comprehension."
There is a road that represents how the Q travel space and time, as well as various people and objects that the Q take turns 'being.'
That said, the Q tend to underestimate human abilities(even if they may fear our potential), so it's possible that the Q gave Voyagers crew a more-dumbed-down-than-necessary version of their reality. At the very least, the Q don't consider humanity capable of perceiving them.
Captain Picard is probably the human who came closest to perceiving what the Q are/what their reality is. In "All Good Things", he closes an anomaly that threatens the human race by co-coordinating three Enterprise crews across three moments in time, and even this seems to be barely scratching the surface of the level at which the Q exist.
The closest we see to a Q in its "natural" form may be when Q chases the Enterprise in the form of a sphere of energy in the pilot episode of TNG, Encounter at Farpoint(at about 11:40 in part 1). However, there's no real evidence that this sphere isn't just another form Q is slipping into for his own amusement.