One way to analyze the problem is to calculate the Lagrangian, and solve for the equation of motion. If you're looking for steady-state solutions (solutions in which the angle of the bead isn't changing), set theta-double-dot to zero and the equilibrium (steady-state) points fall out - either theta equals zero, or theta equals the arc-cosine of g divided by the radius of the circle times omega-squared. So there is the possibility of steady state solutions at points other than theta=zero.
But that still doesn't tell us which points are stable. Stable points are those at which the bead will stay near, even if you nudge the bead a bit; a ball at the bottom of a hill is at a stable point, but a ball at the top of a hill is not - even though all the forces are balanced at the top of the hill, a slight nudge will result in the ball going further and further from the top.
Before we get into that question though, I have a detour to make.