Presumably, you're looking for songs sung when spinning, rather than songs about spinning or spinners.  Again, presumably, you're talking about solo spinning rather than the industrial variety, so the Belfast, Lancashire and Dundee traditions -for example- won't be much use to you.  Mostly, a spinner would sing any popular song of the day that was rhythmically helpful, though there will have been more specific songs; these seem to turn up more often in non-Anglophone traditions where the industrial revolution came later.

It was suggested in a discussion here some time ago that The Frog in the Well, in its early forms, showed strong signs of having been a song used in spinning.  I have no idea if there's any real foundation for that, but you can see staff notation and text modernised from Thomas Ravenscroft's Melismata of 1611 at Greg Lindahl's  The Music of Thomas Ravenscroft:

The Marriage of the Frogge and the Mouse

Malcolm