  A $d$-dimensional, $n$th-order topological insulator or superconductor has
localized eigenmodes at its $(d-n)$-dimensional boundaries ($n\leq d$). In this
work, we apply periodic driving fields to two-dimensional superconductors, and
obtain a wide variety of Floquet second-order topological superconducting
(SOTSC) phases with many Majorana corner modes at both zero and $\pi$
quasienergies. Two distinct Floquet SOTSC phases are found to be separated by
three possible kinds of transformations, i.e., a topological phase transition
due to the closing/reopening of a bulk spectral gap, a topological phase
transition due to the closing/reopening of an edge spectral gap, or an entirely
different phase in which the bulk spectrum is gapless. Thanks to the strong
interplay between driving and intrinsic energy scales of the system, all the
found phases and transitions are highly controllable via tuning a single
hopping parameter of the system. Our discovery not only enriches the possible
forms of Floquet SOTSC phases, but also offers an efficient scheme to generate
many coexisting Majorana zero and $\pi$ corner modes that may find applications
in Floquet quantum computation.
