Joint work with Godofredo Iommi. In previous work with Johansson, Oberg and Pollicott we gave a `conditional variational principle' for the multifractal spectra of Birkhoff averages for certain non-uniformly hyperbolic maps. In this talk we show how a different approach via the thermodynamic formalism for countable state Markov maps can give more information about the regularity of the spectra for continuous potentials. In particular we'll show how this can be used to find discontinuities in the spectrum and where the spectrum is analytic.