Are singers musicians?

The human voice is a musical instrument. Professional singers, particularly operatic, generally require extensive ongoing training in the use of their voice as a musical instrument.
If the definition of musician is one who uses or plays a musical instrument then singers are by definition musicians.

Absolutely yes.

That depends on your definition of musician. Loosely, anyone who makes music, vocal, percussion, or instrumental, is a musician.

If you hold the term musician to a higher standard, then only someone who is musically literate and has developed both the specialized knowledge and skills of music can be called a musician. Certainly, there are literate, skilled, and knowledgeable vocal musicians (Annie Lennox for example), but the average pop singer is not. And by this definition, the musicianship of many pop/rock instrumentalists could also be denied.