Can one define wavefronts for waves travelling on a stretched string? If I have a wave on a string, can any wavefront be defined for such a wave?
And also is it possible to have circularly polarized string waves?
 A: 
If I have a wave on a string, can any wavefront be defined for such a wave? 

In general, a wavefront is defined as a connected set of points in a wave that are all at the same phase at a given time (usually at the phase corresponding to the maximum displacement.)  For a wave traveling in 1-D, the points at which the string is at the same phase are disconnected from each other;  so in some sense, each wavefront consists of a single point.  
This actually makes sense if you think about it.  For a wave traveling in 3-D, the wavefronts are two-dimensional surfaces;  for a wave traveling in 2-D, the wavefronts are one-dimensional curves;  and so for a wave traveling in 1-D, the wavefronts are zero-dimensional points.

And also is it possible to have circularly polarized string waves?

Sure thing.  You have two independent transverse polarizations;  just set up a wave where these two polarizations are 90° out of phase with each other.  The result would be a wave that looks like a helix propagating down the string.  The animation from Wikipedia below was created with electric fields in a circularly polarized light wave in mind;  but the vectors in the animation could equally well represent the displacement of each point of the string from its equilibrium position.

