In this paper, by the moving spheres method, Caffarelli-Silvestre extension formula and blow-up analysis, we study the local behaviour of nonnegative solutions to fractional elliptic equations
\begin{align*} (-\Delta)^{\alpha} u =f(u),~~ x\in \Omega\backslash \Gamma, \end{align*}
where
$0<\alpha <1$,
$\Omega = \mathbb {R}^{N}$ or
$\Omega$ is a smooth bounded domain,
$\Gamma$ is a singular subset of
$\Omega$ with fractional capacity zero,
$f(t)$ is locally bounded and positive for
$t\in [0,\,\infty )$, and
$f(t)/t^{({N+2\alpha })/({N-2\alpha })}$ is nonincreasing in
$t$ for large
$t$, rather than for every
$t>0$. Our main result is that the solutions satisfy the estimate
\begin{align*} f(u(x))/ u(x)\leq C d(x,\Gamma)^{{-}2\alpha}. \end{align*}
This estimate is new even for
$\Gamma =\{0\}$. As applications, we derive the spherical Harnack inequality, asymptotic symmetry, cylindrical symmetry of the solutions.