  We consider the inviscid unsteady Prandtl system in two dimensions, motivated
by the fact that it should model to leading order separation and singularity
formation for the original viscous system. We give a sharp expression for the
maximal time of existence of regular solutions, showing that singularities only
happen at the boundary or on the set of zero vorticity, and that they
correspond to boundary layer separation. We then exhibit new Lagrangian
formulae for backward self-similar profiles, and study them also with a
different approach that was initiated by Elliott-Smith-Cowley and
Cassel-Smith-Walker. One particular profile is at the heart of the so-called
Van-Dommelen and Shen singularity, and we prove its generic appearance (that
is, for an open and dense set of blow-up solutions) for any prescribed Eulerian
outer flow. We comment on the connexion between these results and the full
viscous Prandtl system. This paper combines ideas for transport equations, such
as Lagrangian coordinates and incompressibility, and for singularity formation,
such as self-similarity and renormalisation, in a novel manner, and designs a
new way to study singularities for quasilinear transport equations.
