TLDW
A paper, by philosopher of physics John Norton, describes a scenario in which a ball is placed at the apex of a perfectly smooth and frictionless dome. The paper argues that, according to Newton's laws, the ball will spontaneously start rolling down the dome at some random time, without any external force acting on it. Norton's claim suggests that Newtonian mechanics may not a deterministic theory or that there are multiple possible interpretations.
https://sites.pitt.edu/~jdnorton/papers/DomePSA2006.pdf
The discussion about whether Newton's first law actually states that the forces are first causes for the motion is very interesting. However, I do not find the comparison with the harmonic oscillator appropriate : vanishing instantaneous acceleration at one point in time (like in the harmonic oscillator) is not the same thing as an inertial motion.