The distribution for a coin is uniform over the event space [heads,tails]; not over the angle of rotation it lands on. You can have continuous or discrete event spaces.
If you have something shaped like a gömböc[1] you can have a "coin" that lands on heads ~100% of the time, except for the whole not-actually-being-a-coin part.
Because physics - it's not possible to bias a rigid body so that it rotates with non-constant angular speed when flipped, as long as air resistance can be neglected, and that means that a fair flip gives 50/50 odds as to what side you catch it on. (Edit: clarity)
If other stuff is going on, like you're letting it bounce or something, then it depends on the particulars. It's rather easy to load a die, for instance.
From that perspective the uniform prior I use is reasonable (though non-committal and fairly uninformative), a triangular distribution with its peak at 50% (http://en.wikipedia.org/wiki/Triangular_distribution), the prior I have after updating tails and then heads, a normal distribution, or using the beta distribution with an equal number of both parameters.
Honestly you may decide there's a fair chance the coin is heads biased so you choose a prior that has most of the mass of the probability above 50%. As long as you have a reason, it's reasonable. You don't have to worry too much that you've chosen the optimal prior since more data makes equals of priors in the long run.
The only terrible thing you can do, is pick a prior that absolutely excludes certain hypotheses with 0% likelihood. No amount of data can overcome that.
That may conflict with your sensibilities if you expect very strange coins, but if the priors aren't too crazily bad, if you have enough time for a few more coin flips, and the coin isn't particularly brittle, what's an order of magnitude between friends?
Edit: And for formal mathy purposes, ask someone else :)