In this talk, I will present a recent study on  Burgers-Fisher-KPP equation with singular slow/fast diffusion and singular/regular convection in the form of $u_t-D\Delta u^m+\alpha(u^p)_x=f(u)$ with $m,\,p>0$, focusing on the existence, non-existence, regularity and stability of traveling waves. The values of m and p essentially affect the existence/non-existence of regular/sharp traveling waves as well as their regularity.   By combining phase-plane analysis and variational techniques, we obtain a complete classification of existence/non-existence of regular/sharp traveling waves related to m and p. In the singular regimes with 0<p<1 or 0<m<1, where the convection or diffusion exhibit strong singularity at u=0, we introduce a change of variables to overcome the singularity, thereby deriving existence/non-existence results characterized by the minimal convection coefficient. Finally, for the case of slow diffusion $m>1$ with convex convection p>1, we prove the stability of non-critical traveling waves via $L^{1}$-weighted energy method. This is a joint work with Zhuangzhuang Wang, Rui Huang, Zejia Wang and Wenhuan Liang.