How many Slip rings in synchronous motor?

If the synchronous motor is single phase then there are two slip rings & if this motor is three phase so the slip rings are three in number.

Correction;

I have never seen a single phase synchronous motor. It would cost more that it was worth.

A synchronous motor has a separately excited field. If the excitation comes from a stationary DC source it has 2 slip (collector) rings.

A brushless induction motor has no slip rings because the exciter armature rotates and so do the rectifiers.

A permanent magnet motor, used with variable frequency drives, is another type of synchronous motor that has no slip rings.

A three phase motor with 3 slip rings is a Wound Rotor motor. Wound rotor motors are variable speed motors that were used for such applications as bridges and cranes before variable speed drives.