Answer

• Finite verb forms show tense, person and number (I go, she goes, we went, etc.):

She was waiting in the room before he came in.

Does your brother know my brother?

The night before he had to leave, they sat on the small sofa in the living-room and looked at old family photos.

Aren’t you a bit late?

• Non-finite verb forms do not show tense, person or number. Typically they are infinitive forms with and without to (e.g. to go, go), -ing forms and -ed forms (e.g. going, gone):

She tiptoed round the house so as not to wake anyone.

You need to paint the whole cupboard, starting from the bottom.

So for the given sentence, 

• Singing is his past time

the finite verb would be: is.