We study a singular elliptic problem driven by a mixed local-nonlocal operator of the form
\begin{equation*}
\begin{aligned}
-\Delta_p u + (-\Delta_q)^s u &= \frac{\lambda}{u^{\delta}} + u^r \text{ in } \Omega\\
u>0 \text{ in } \Omega,\ u &=0 \text{ in } \mathbb{R}^N \setminus \Omega
\end{aligned
\end{equation*}
where $p > sq$, $0<\delta<1$ and $\lambda > 0$ is a parameter. The nonlinearity exhibits a singular power-type behavior near zero and displays at most a critical growth at infinity. We establish a global multiplicity result with respect to the parameter $\lambda$ by identifying a sharp threshold that separates existence, non-existence, and multiplicity regimes. We also derive a Hopf-type strong comparison principle adapted to this nonlinear setting, which provides the main analytical tool for the global multiplicity result. Additionally, we establish qualitative properties of solutions that are crucial for the variational analysis, including a uniform $L^{\infty}$-estimate and a Sobolev versus H\"older local minimizer result. The analytical tools developed here are of independent interest and apply to a broader class of mixed local-nonlocal equations.