Can you cover a non-singular algebraic variety by pairwise disjoint singular closed subvarieties? Varieties are over an algebraically closed field of characteristic other than $2$ and $3$.

In characteristic $2$ and $3$ this is possible via quasi-elliptic surfaces as pointed out by abx.

If we allow locally closed subvarieties then $\mathbb{A}^2$ can be covered by cuspidal cubics $(y-t)^2=x^3$ (whenever two cubics intersect remove the finitely many intersection points from one of them).