A positive charge Q is uniformly distributed along a circular ring of radius R. A small test charge q is placed at the centre of the ring in figure. Then

The positive charge Q is uniformly distributed over the cicumference of the ring . At the centre of the ring electric field is zero but at any other point inside the ring and in the plane of the ring , resultant electric field is not zero and directed towards the centre of the ring .

 So, the effect of electric field on charge q due to the positive charge Q is zero.

 Now, the only governing factor is the attractive and repulsive forces between charges (Q and q) there are two cases arise. 

Case l When charge q> 0, i.e. q is a positive charge, they creates a repulsive force between charge q and Q.

 The repulsive forces of charge Q from all around the charge q will push it towards the centre if it is displaced from the centre of the ring. 

Case II When charge q < 0, i.e. q is a negative charge, then there is an attractive force between charge Q and q.

 If q is shifted from the centre, then the positive charges nearer to this charge will attract it towards itself and charge q will never return to the centre. 

q is in stable equilibrium in the plane of the ring .

Electric field , produced by Q ,at any point on the axis of the ring is directed away from the centre of the ring . So if q$<0 and the charge q is slightly displaced from the centre and released,$ it will be pushed back towards  the centre . So q is in stable equilibrium along the axis of the ring and for small displacement q will perform SHM.