  We extend many results on Selmer groups for elliptic curves and modular forms
to the non-ordinary setting. More precisely, we study the signed Selmer groups
defined using the machinery of Wach modules over $\mathbf{Z}_p$-cyclotomic
extensions. First, we provide a definition of residual and non-primitive Selmer
groups at non-ordinary primes. This allows us to extend techniques developed by
Greenberg (for $p$-ordinary elliptic curves) and Kim ($p$-supersingular
elliptic curves) to show that if two $p$-non-ordinary modular forms are
congruent to each other, then the Iwasawa invariants of their signed Selmer
groups are related in an explicit manner. Our results have several
applications. First of all, this allows us to relate the parity of the analytic
ranks of such modular forms generalizing a recent result of the first-named
author for $p$-supersingular elliptic curves. Second, we can prove a Kida-type
formula for the signed Selmer groups generalizing results of Pollack and
Weston.
