  We establish finite-time singularity formation for $C^{1,\alpha}$ solutions
to the Boussinesq system that are compactly supported on $\mathbb{R}^2$ and
infinitely smooth except in the radial direction at the origin. The solutions
are smooth in the angular variable at the blow-up point, which was a
fundamental obstruction in previous works. This is done by exploiting a
second-order effect, related to the classical Rayleigh--B\'enard instability,
that overcomes the regularizing effect of transport. A similar result is
established for the 3d Euler system based on the Taylor--Couette instability.
