Is there a physical upper limit on how fast a physical object can rotate? Objects can not travel faster than the speed of light.
Angular velocity is $\omega = \frac{s \times v}{|s|^2}$
It seems to me that for infinitely small objects (like electrons) there is no limit on how fast objects can rotate.
It seems wrong to me that it is theoretically possible to generate more magnetic effects and similar by rotating than by linear movement though.
I recall that sufficiently fast rotating black holes can form naked singularities. If rotation is limited to below that amount then that might be a limit on how fast things can rotate?
 A: The underlying level of nature is quantum mechanical, all classical frameworks emerge from this underlying level. Hopefully this will be true of general relativity too. Already string theories can accommodate both quantization of gravity and the structure of the standard model of physics with no definitive model yet. 
Therefore the speed at which an object can rotate will be limited by the electromagnetic forces holding the object together against the mechanical centrifugal forces, and will depend on the atomic structure of the object. No need of speed of light, already if you rotate a piece of dough the outer levels will fly away :). One would have to calculate the forces for a specific size steel ball , for example, and get a limit for that ball. It will answer how fast must it rotate for the surface atoms to split off due to centrifugal forces. 
Now as far as black holes go, they are a classical construct, and singularities are a classical theory effect. Think of the 1/r^2 of the coulomb force which leads to a singularity at r=0. Quantization took care of that. When gravity will be quantized, I expect that the same will happen with its singularities.

It seems to me that for infinitely small objects (like electrons) there is no limit on how fast objects can rotate.

The electron is a quantum mechanical entity, an elementary particle in the standard model,  it is a point particle, there is no extent in space to describe a rotation . The  spin  attributed to the electron is a quantum number necessary to describe the interaction of electrons with matter, so that angular momentum is conserved in all electron interactions with other particles. It is an observational/experimental   fact.
