  In this article we consider the discretely self-similar singular solutions of
the Euler equations, and the possible velocity profiles concerned not only have
decaying spatial asymptotics, but also have unconventional non-decaying
asymptotics. By relying on the local energy inequality of the velocity profiles
and the bootstrapping method, we prove some nonexistence results and show the
energy behavior of the possible nontrivial velocity profiles. For the case with
non-decaying asymptotics, the needed representation formula of the pressure
profile in terms of velocity profiles is also given and justified.
