  We investigate the Friedel oscillations of the local density of states (LDOS)
induced by a single impurity with both a spin-independent potential and an
exchange coupling to the electrons in altermagnets and unconventional $p$-wave
magnets. We identify features that make the Friedel oscillations and
magnetization distinct from other materials with nontrivial spin texture such
as Rashba metals. Because time-reversal symmetry is broken in altermagnets,
both magnetic and nonmagnetic impurities lead to local magnetization with the
spatial pattern that reflects the symmetry of the altermagnetic splitting. The
period of the corresponding oscillations provides an alternative way to
quantify the altermagnetic spin splitting and the shape of the altermagnetic
bands. The LDOS pattern in $p$-wave magnets, which respect combined
time-reversal and translation symmetries, is rich. It reveals anisotropy
related directly to the spin splitting, but surprisingly also features LDOS
oscillations with a doubled period in the proximity of the impurity. The latter
effect is also observed in a Rashba metal with an exchange field and originates
from the interplay of propagating and evanescent waves. The obtained results
are instrumental for investigating altermagnets and unconventional $p$-wave
magnets and impurity effects via tunneling probes.
