We consider estimation of regression models whose error terms follow a finite mixture of scale mixtures of skew-normal (SMSN) distributions, a rich class of distributions that contains the skew-normal, skew-t, skew-slash and skew-contaminated normal distributions as proper elements. This approach allows us to model data with great flexibility, accommodating simultaneously multimodality, skewness and heavy tails. We developed a simple EM-type algorithm to perform maximum likelihood (ML) inference of the parameters of the proposed model with closed-form expression at the E-step. Furthermore, the standard errors of the ML estimates can be obtained as a byproduct. The practical utility of the new method is illustrated with the analysis of real dataset and several simulation studies. The proposed algorithm and methods are implemented in the R package FMsmsnReg().