  In this paper, we investigate sharp singular Trudinger-Moser inequalities
involving the anisotropic Dirichlet norm $\left(\int_{\Omega}F^{N}(\nabla
u)\;\mathrm{d}x\right)^{\frac{1}{N}}$ in the Sobolev-type space
$D^{N,q}(\mathbb{R}^{N})$, $q\geq 1$, here
$F:\mathbb{R}^{N}\rightarrow[0,+\infty)$ is a convex function of class
$C^{2}(\mathbb{R}^{N}\setminus\{0\})$, which is even and positively homogeneous
of degree 1, its polar $F^{0}$ represents a Finsler metric on $\mathbb{R}^{N}$.
Combing with the connection between convex symmetrization and Schwarz
symmetrization, we will establish anisotropic singular Trudinger-Moser
inequalities and discuss their sharpness under several different situations,
including the case $\|F(\nabla u)\|_{N}\leq 1$, the case $\|F(\nabla
u)\|_{N}^{a}+\|u\|_{q}^{b}\leq 1$, and whether they are associated with exact
growth.
