Partially linear single-index models represent a versatile tool to capture the relationship between response variables and possibly high-dimensional covariate vectors. The approximation of the response is given by the sum of a linear term and of a nonparametric link function of a second linear combination of covariates, usually called the index. This approximation is defined with respect to a loss function which characterizes a feature of the conditional law of the response given the covariates such as the conditional mean or the conditional median. In this paper we consider a general family of loss functions and investigate the corresponding partially linear single-index regression models, including mean, quantile, expectile and robust regressions. Except for imposing some moments to be finite, the conditional law of the error term is allowed to be general. For the inference, we adopt the empirical likelihood (EL) approach based on a class of moment conditions in which we plug-in estimates of the nuisance link function. We show the asymptotic pivotality of the likelihood ratio under weak high-level conditions. A simple data-driven choice of the tuning parameter for the estimation of the link function.