The usual units are J/K, however the only reason temperature has different units than energy is because we didn't used to know temperature was just the average kinetic energy of atoms/molecules. Hence, using a more reasonable system, the units would be J/J which just cancel.
I get that we're stuck with having different units for temperature now, just as we're stuck with the inconvenient convention that electrons are negative and protons positive. But since we already have a constant in the definition of entropy anyway, why haven't we defined it in a way as to convert the temperature units to energy and so make entropy dimensionless?
Or, better yet, why not, for the purposes of thermodynamics problems, just use subsume the Boltzmann constant into the units for temperature in the first place? That would make certainly make for a less messy system of units and make the units of both temperature and entropy better match the definitions of the quantities. I get that we can't just switch the temperature unit system for everyone, but I'm rather surprised scientists working in thermodynamics (thermodynamicists?) haven't adopted that unit convention within the field.
