  In this paper, we use the method of blow-up analysis and capacity estimate to
derive the singular Trudinger--Moser inequality involving
$N$-Finsler--Laplacian and $L^{p}$ norm, precisely, for any $p>1$,
$0\leq\gamma<\gamma_{1}:= \inf\limits_{u\in W^{1, N}_{0}(\Omega)\backslash
\{0\}}\frac{\int_{\Omega}F^{N}(\nabla u)dx}{\| u\|_p^N}$ and $0\leq\beta<N$, we
have \begin{align} \sup_{u\in W_{0}^{1,N}(\Omega),\;\int_{\Omega}F^{N}(\nabla
u)dx-\gamma\|
u\|_p^N\leq1}\int_{\Omega}\frac{e^{\lambda_{N}(1-\frac{\beta}{N})\lvert
u\rvert^{\frac{N}{N-1}}}}{F^{o}(x)^{\beta}}\;\mathrm{d}x<+\infty\notag,
\end{align} where $\lambda_{N}=N^{\frac{N}{N-1}} \kappa_{N}^{\frac{1}{N-1}}$
and $\kappa_{N}$ is the volume of a unit Wulff ball in $\mathbb{R}^N$,
moreover, extremal functions for the inequality are also obtained. When
$F=\lvert\cdot\rvert$ and $p=N$, we can obtain the singular version of Tintarev
type inequality by the obove inequality, namely, for any
$0\leq\alpha<\alpha_{1}(\Omega):=\inf\limits_{u\in W^{1,
N}_{0}(\Omega)\backslash \{0\}}\frac{\int_{\Omega}|\nabla u|^Ndx}{\| u\|_N^N}$
and $0\leq\beta<N$, it holds $$ \sup_{u\in
W_{0}^{1,N}(\Omega),\;\int_{\Omega}\lvert\nabla
u\rvert^{N}\;\mathrm{d}x-\alpha\|u\|_{N}^{N}\leq1}\int_{\Omega}\frac{e^{\alpha_{N}(1-\frac{\beta}{N})\lvert
u\rvert^{\frac{N}{N-1}}}}{\lvert x\rvert^{\beta}}\;\mathrm{d}x<+\infty, $$
where $\alpha_{N}:=N^{\frac{N}{N-1}}\omega_{N}^{\frac{1}{N-1}}$ and $
\omega_{N}$ is the volume of unit ball in $\mathbb{R}^{N}$. Our results extend
many well-known Trudinger--Moser type inequalities to more general setting.
