Summary: In this paper, we introduce a flexible class of regression models for counts with high-inflation of zeros that cannot be predicted by the Poisson, the zero-inflated Poisson, the negative binomial and the Poisson-inverse Gaussian regression models. Our proposed flexible regression models are based on a class of zero-inflated mixed Poisson distributions and contain the zero-inflated negative binomial (ZINB) and the zero-inflated Poisson-inverse Gaussian (ZIPIG) distributions, as particular cases, among others. We consider regression structures for the mean, the dispersion, and the zero-inflation parameters. Consequently, we generalize existing models, such as the ZINB regression (with non-varying dispersion), and also open the possibility of introducing new models, such as the ZIPIG and the zero-inflated generalized hyperbolic secant regressions. We propose an Expectation-Maximization (in short EM) algorithm for estimating the parameters and the associated information matrix. Simulation results are presented to compare the finite-sample performance of our proposed EM-algorithm with a direct maximization of the log-likelihood function based on the GAMLSS approach. These simulated results show some advantages of our EM-algorithm concerning the GAMLSS proposal. We also discuss a measure of influence based on the EM approach and propose simulated envelopes for checking the adequacy of our zero-inflated regression models. An empirical application, about the number of roots produced by 270 micropropagated shoots of the columnar apple cultivar Trajan, illustrates the usefulness of the proposed class of regression models for dealing with count data presenting high-inflation of zeros and shows that one cannot use the GAMLSS approach in some practical situations due to numerical problems.