  We use the Higgs mechanism to investigate connections between higher-rank
symmetric $U(1)$ gauge theories and gapped fracton phases. We define two
classes of rank-2 symmetric $U(1)$ gauge theories: the $(m,n)$ scalar and
vector charge theories, for integer $m$ and $n$, which respect the symmetry of
the square (cubic) lattice in two (three) spatial dimensions. We further
provide local lattice rotor models whose low energy dynamics are described by
these theories. We then describe in detail the Higgs phases obtained when the
$U(1)$ gauge symmetry is spontaneously broken to a discrete subgroup. A subset
of the scalar charge theories indeed have X-cube fracton order as their Higgs
phase, although we find that this can only occur if the continuum higher rank
gauge theory breaks continuous spatial rotational symmetry. However, not all
higher rank gauge theories have fractonic Higgs phases; other Higgs phases
possess conventional topological order. Nevertheless, they yield interesting
novel exactly solvable models of conventional topological order, somewhat
reminiscent of the color code models in both two and three spatial dimensions.
We also investigate phase transitions in these models and find a possible
direct phase transition between four copies of $\mathbb{Z}_2$ gauge theory in
three spatial dimensions and X-cube fracton order.
