  Superconductivity in the recently discovered altermagnetic materials hosts
large prospects for both fundamental physics and technological applications. In
this work we show that a characteristic spin-sublattice locking in altermagnets
puts severe constraints on possible superconducting pairing. In particular, we
uncover that the most common form of superconductivity, uniform $s$-wave
spin-singlet pairing is not possible to achieve in altermagnets. Considering an
effective model for a $d_{x^2-y^2}$-wave altermagnet on a square lattice, we
instead find that the most likely forms of spin-singlet pairing have
$d_{x^2-y^2}$- or extended $s$-wave symmetry. We also find that the simplest
form of equal-spin-triplet $p$-wave pairing is not allowed, but it can only
exist as a mixed-spin-triplet $p$-wave state. We verify these constraints on
pairing within an interaction-induced model of altermagnetism, where we also
establish their validity for finite-momentum pairing. Additionally we discuss
the possible pairing symmetries for odd-frequency superconducting pairing. Due
to the generality of our results, they are applicable to both intrinsic
superconductivity and proximity-induced superconductivity in
altermagnet-superconductor hybrid junctions.
