I've experienced wobbles on bikes and motorcycles, and they can be scary. They are a kind of gyroscopic thing in which destabilizes the wheel when it tries to move to a different phase. Wheels spin smoothly in equilibrium when they are perfectly vertical or horizontal, if they are anywhere in between these positions they try to push back to equilibrium, in certain situations they fluctuate between phases. The effect amplifies itself as the wobble increases. To minimize the potential for this problem, keeping the wheels true and in balance is important. Another possible cause is fork/frame misalignment, or a loose headset.
You can balance the front wheel with a cycle computer magnet which attaches to a spoke. Normally when you spin the wheel, it will stop with the heaviest side down. Attach the magnet to a spoke opposite to the heavy side. Moving the magnet closer or further away from the hub can fine tune the balance. When the wheel is balanced, it should stop turning in random positions. Heavy grease in the headset and tightening the bearings enough to add a little resistance when turning the fork can help. Motorcycles use dampeners and/or stabilizers for the same effect.