In classical Newtonian physics, singularities exist, but only in frictionless systems. One example is the driven oscillator. Another more complex one the https:

In classical Newtonian physics, singularities exist, but only in frictionless systems. One example is the driven oscillator. Another more complex one the https://en.wikipedia.org/wiki/Painlev%C3%A9_conjecture https://en.wikipedia.org/wiki/Painlev%C3%A9_conjecture . However, as soon as friction is taken into account, the singularities (e.g., infinitely high amplitudes in the oscillator) are damped. Fluids without internal friction or viscosity are described by the Euler equations. Therefore, singularities (finite time blowups) occur there, which has recently been shown with a computer assisted proof: https://www.quantamagazine.org/computer-helps-prove-long-sought-fluid-equation-singularity-20221116/ https://www.quantamagazine.org/computer-helps-prove-long-sou... The Navier-Stokes equations are the Euler equations plus friction/viscosity: As a physicist, I do not expect singularities there, since energy is always lost due to friction.