It was a lot of fun to derive the solution for the cubic and quartic equation 22 years ago, when I was a high school kid, but I haven't really used it afterwards. I think the same is true for most physicists I met: they didn't use the complicated formulae, especially not the quartic one.
Much more generally, the situations in which it would be useful are extremely rare for several reasons:
the situations in which the functions are truncated to a cubic or quartic polynomial are extremely rare
known corrections ultimately modify the cubic or quartic polynomial to a more general function
the position of roots is usually not the key interesting quantity
the truly interesting and invariant quantities are functions of the roots that can be expressed more easily in terms of the coefficients
quite typically, physical problems wouldn't lead to a cubic or quartic equation for one variable but to a set of such equations for many variables which can't be solved in radicals
in the interesting cases when a cubic or quartic equation has to be solved, it reduces to a linear or quadratic equation, anyway.
To give some examples, one may consider the most general classical potential for a scalar field which produces a renormalizable theory. This is given by at most a 4th order polynomial of the scalar field. For a general potential, the zeros of such potential would be roots of the polynomial.
However, zeros of a potential are not too interesting: much more interesting are its stationary points, i.e. extrema, which are the solutions of the cubic equation. Moreover, the coefficients in the polynomial remain more important than the positions of the minima - and observable quantities may be expressed as their functions.
Also, the canonical shapes of the potential have a $Z_2$ or a related symmetry that effectively reduces the degree of the equation to half of it. Moreover, generally, the theories may include several fields which produces a set of higher-order equations that usually can't be solved in radicals. Equally importantly, quantum corrections add non-polynomial corrections to the potential, anyway.
So the set of physical situations in which the problem could be solved using the nontrivial formulae is limited for all the reasons above, and even when the problem may be solved, we usually don't learn anything "qualitative" about the system. In this sense, the existence of the solution for cubic and quartic solutions is a mathematical curiosity that doesn't affect a physicist's life.