Since 2000
120. Moffatt, H.K. (2000) Euler's disk and its finite-time singularity. Nature 404, 833-834.
Euler's diskis a toy disk that can be easily set in precessional spinning motion on a table. It is interesting from a mechanical point of view because it exhibits a finite-time singularity, viz. an infinite precessional angular velocity within a finite time, about 100 seconds. This infinity is in practise resolved in a matter of milliseconds. I sought to explain this behaviour through analysing the viscous dissipation in the thin layer of air trapped between the disc and the table. Other dissipative mechanisms are undoubtedly present, but only this one has been analysed in detail.
The 'finite-time singularity problem' in fluid mechanics relates to smoothness of solutions of the Navier-Stokes equations: can a singularity of vorticity appear at finite time starting from smooth initial conditions? We don't yet know the answer to this very fundamental question. In the meantime, it is useful to have the Euler disk as a table-top model of finite-time-singularity behaviour.
I bought this toy in 1999 as a Christmas present for a grandchild; it turned out to be too interesting to give away!