I think the paper you want is B. Halpern, "Strange Billiard Tables." Transactions of the AMS Vol 232, 1977.

It looks like he constructs a $C^2$ catastrophe (with collision points on the unit circle, but an irregularly shaped table passing through those points) and rules out a $C^3$ catastrophe.

I think the paper you want is B. Halpern, "Strange Billiard Tables." Transactions of the AMS Vol 232, 1977.

Thanks to Carl for pointing out that Halpern considers tables with the additional condition of nonvanishing curvature. He constructs a $C^2$ catastrophe (with collision points on the unit circle, but an irregularly shaped table passing through those points) and rules out a $C^3$ catastrophe.

I believe that without the condition of nonvanishing curvature, Halpern's construction can be adapted to produce a smooth catastrophe with infinitely many collision points on $\exp(-1/x^2)$ approaching the origin. However, I haven't verified that the result can be made smooth.