We analyze the moment of inertia $\MI(S)$, relative to the center of gravity, of finite plane lattice sets $S$. We classify these sets according to their roundness: a set $S$ is rounder than a set $T$ if $\MI(S) < \MI(T)$. We show that roundest sets of a given size are strongly convex in the discrete sense. Moreover, we introduce the notion of quasi-discs and show that roundest sets are quasi-discs. We use weakly unimodal partitions and an inequality for the radius to make a table of roundest discrete sets up to size $40$. Surprisingly, it turns out that the radius of the smallest disc containing a roundest discrete set $S$ is not necessarily the radius of $S$ as a quasi-disc.