Is jack and the beanstalk a nursery rhyme or a fairy tale?

No.

A nursery rhyme is a short verse that is often quoted to children.

Example: Humpty Dumpty sat on a wall.

Humpty Dumpty had a great fall.

All the kings horses and all the kings men,

Could not put Humpty Dumpty together again.

A fairy tale is a fantasy story. In modern times they are geared more towards children, but that was not always the case. Examples of Fairy Tales include 'Snow White', 'Cinderella', 'The Little Mermaid', and 'Sleeping Beauty'. These stories are actually very old and were not written much in the style that companies like Disney have changed them into.