Multivariate extreme value theory mostly focuses on asymptotic dependence, where the probability of observing a large value in one of the variables is of the same order as that of observing a large value in all variables simultaneously. There is growing evidence, however, that asymptotic independence prevails in many data sets. Available statistical methodology in this setting is scarce and not well understood theoretically. We revisit non-parametric estimation of bivariate tail dependence and introduce rank-based M-estimators for parametric models that may include both asymptotic dependence and asymptotic independence, without requiring prior knowledge on which of the two regimes applies. We further show how the method can be leveraged to obtain parametric estimators in spatial tail models. All the estimators are proved to be asymptotically normal under minimal regularity conditions. The methodology is illustrated through an application to extreme rainfall data.