  We study the class of transversal submanifolds. We characterize their
blow-ups at transversal points and prove a negligibility theorem for their
"generalized characteristic set", with respect to the Carnot-Carath\'eodory
Hausdorff measure. This set is made by all points of non-maximal degree.
Observing that C^1 submanifolds in Carnot groups are generically transversal,
the previous results prove that the "intrinsic measure" of C^1 submanifolds is
generically equivalent to their Carnot-Carath\'eodory Hausdorff measure. As a
result, the restriction of this Hausdorff measure to the submanifold can be
replaced by a more manageable integral formula, that should be seen as a
"sub-Riemannian mass". Another consequence of these results is an explicit
formula, only depending on the embedding of the submanifold, that computes the
Carnot-Carath\'eodory Hausdorff dimension of C^1 transversal submanifolds.
