On the degrees of freedom count on singular phase space submanifolds

I discuss singular loci in the phase spaces of theories which lack globally well-defined numbers of dynamical modes. This is a topic which appears quite often in the recent literature on modified gravity. In particular, there were discussions about $R^2$ gravity around Minkowski space. It is a relatively simple case, and still there were some confusions. It clearly shows that one should be very accurate when trying to understand a potentially problematic theory through perturbations around a simply looking background. At the same time, many modern teleparallel approaches are laden with even more severe issues. Therefore, it is a topic which is certainly worth carefully thinking about.