A particularly simple example is Norton's dome, with height given as a function of radial distance on the surface of the dome by
$h = \frac{2}{3g}r^{3/2}$
where $g$ is the gravitational constant near the surface of the earth. The dome has a curvature singularity at the apex. And, if we model a mass at the ($r=0$) apex of this dome with zero velocity, we find that Newton's equation does not have a unique solution; the mass can "fall" at any arbitrary time $t$ for no reason at all.
Norton's paper about the dome: http://philsci-archive.pitt.edu/2943/
A helpful reply: http://philsci-archive.pitt.edu/3195/
