  We discuss existence, non-uniqueness and regularity of one- and two-sided
solutions of initial value problems for scalar quasi-linear ordinary
differential equations where the initial condition corresponds to an impasse
point of the equation. With a differential geometric approach, we reduce the
problem to questions in dynamical systems theory. As an application, we discuss
in detail second-order equations of the form $g(x)u''=f(x,u,u')$ with an
initial condition imposed at a simple zero of $g$. This generalises results by
Liang and also makes them more transparent via our geometric approach.
