We consider the application of multilevel Monte Carlo methods to elliptic PDEs with random coefficients. We focus on models of the random coefficient that lack uniform ellipticity and boundedness with respect to the random parameter, and that only have limited spatial regularity. We extend the finite element error analysis for this type of equation, carried out in Charrier et al. (SIAM J Numer Anal, 2013), to more difficult problems, posed on non-smooth domains and with discontinuities in the coefficient. For this wider class of model problem, we prove convergence of the multilevel Monte Carlo algorithm for estimating any bounded, linear functional and any continuously Fréchet differentiable non-linear functional of the solution. We further improve the performance of the multilevel estimator by introducing level dependent truncations of the Karhunen-Loève expansion of the random coefficient. Numerical results complete the paper.