  In the setting of Carnot groups, we propose an approach of taming
singularities to get coercive inequalities. To this end, we develop a technique
to introduce natural singularities in the energy function $U$ in order to force
one of the coercivity conditions. In particular, we explore explicit
constructions of probability measures on Carnot groups which secure Poincar\'e
and even Logarithmic Sobolev inequalities. As applications, we get analogues of
the Dyson-Ornstein-Uhlenbeck model on the Heisenberg group and obtain results
on the discreteness of the spectrum of related Markov generators.
