I like to work with the HLL and related solvers because of their robustness and simple implementation. The Roe solver is another good choice but the entropy fix problem is a turn off. So, I like to know your own favourite solver(s) and why.
I recommend the AUSM+ scheme by Liou (see Liou, J. Comput. Phys. 1996), which has the advantages of both flux-vector and flux-difference splitting schemes. It is only defined for Euler equations (can be extended to Navier-Stokes and chemical reactions if needed), not for transport or Burgers equations.