  This paper studies the existence of $2$-torsion in instanton Floer homology
with $\mathbb{Z}$ coefficients for closed $3$-manifolds and singular knots.
First, we show that the non-existence of $2$-torsion in the framed instanton
Floer homology $I^\sharp(S_n^3(K);\mathbb{Z})$ of any nonzero integral
$n$-surgery along a knot $K$ in $S^3$ would imply that $K$ is fibered. Also, we
show that $I^\sharp(S_{r}^3(K);\mathbb{Z})$ for any nontrivial $K$ with
$r=1,1/2,1/4$ always has $2$-torsion. These two results indicate that the
existence of $2$-torsion is expected to be a generic phenomenon for Dehn
surgeries along knots. Second, we show that for genus-one knots with nontrivial
Alexander polynomials and for unknotting-number-one knots, the unreduced
singular instanton knot homology $I^\sharp(S^3,K;\mathbb{Z})$ always has
$2$-torsion. Finally, some crucial lemmas that help us demonstrate the
existence of $2$-torsion are motivated by analogous results in Heegaard Floer
theory, which may be of independent interest. In particular, we show that, for
a knot $K$ in $S^3$, if there is a nonzero rational number $r$ such that the
dual knot $\widetilde{K}_r$ inside $S^3_r(K)$ is Floer simple, then $S^3_r(K)$
must be an L-space and $K$ must be an L-space knot.
