Can you electrolyse a single H2O molecule?
In a traditional experimental setup this would never happen, as there are always several water molecules. But lets just think about it for the sake of it.

If you wanted to design an experiment where you only have one water molecule, you would have to get two electrodes extremely close together. Lets say you find a way to levitate the molecule between the two electrodes. You might be able to then apply a voltage and tear the molecule apart, forming one H$_2$(g) molecule at the cathode. The oxygen radical would then most likely immediately react with the anode to which it is attracted by the Lorentz force and form an oxide. 

So the result of electrolysing several single molecules of water might result in gaseous hydrogen and no oxygen, but a buildup of oxide at the anode.

However, having two electrodes so close together would probably create a series of problems. Not only to hold the single water molecule in place (which could be achieved by optical levitation) but because quantum mechanical effects such as tunneling currents would occur. Probably there are several more complications that would by all means make it nearly impossible to actually observe this. The knowledge gain derived from such an experiment would likely not justify the effort and expenses. 

EDIT

I just revisited this question and thought about it again:
it's not absolutely necessary that the two H atoms would form an H$_2$ molecule. Depending on the way the cathodes are formed, the electric field might not pull the charged particles together, but to a collision with something else in the vicinity; the cathode or any ambient gas. 
Furthermore, my notion above that "H$_2$(g)" would form is a bit nonsensical, as a single molecule cannot really be attributed a state of matter (i.e., being gaseous in this case).