  Let $G=(V,E)$ be a matching-covered graph and $X$ be an edge set of $G$. $X$
is said to be feasible if there exist two perfect matchings $M_1$ and $M_2$ in
$G$ such that $|M_1\cap X|\not \equiv|M_2\cap X|\ (\mbox{mod } 2)$. For any
$V_0\subseteq V$, $X$ is said to be switching-equivalent to $X\oplus
\nabla_G(V_0)$, where $\nabla_G(V_0)$ is the set of edges in $G$ each of which
has exactly one end in $V_0$ and $A \oplus B$ is the symmetric difference of
two sets $A$ and $B$. Lukot'ka and Rollov\'a showed that when $G$ is regular
and bipartite, $X$ is non-feasible if and only if $X$ is switching-equivalent
to $\emptyset$. This article extends Lukot'ka and Rollov\'a's result by showing
that this conclusion holds as long as $G$ is matching-covered and bipartite.
This article also studies matching-covered graphs $G$ whose non-feasible edge
sets are switching-equivalent to $\emptyset$ or $E$ and partially characterizes
these matching-covered graphs in terms of their ear decompositions. Another aim
of this article is to construct infinite many $r$-connected and $r$-regular
graphs of class 1 containing non-feasible edge sets not switching-equivalent to
either $\emptyset$ or $E$ for an arbitrary integer $r$ with $r\ge 3$, which
provides negative answers to problems asked by Lukot'ka and Rollov\'a and He,
et al respectively.
