Evaluation formulas for Tornheim and Euler type double series

We give closed-form evaluation formulas for the real and imaginary parts of the series $\sum_{m,n=1}^{\infty}\frac{e^{2\pi i\left( mx-ny\right) }} {m^{p}n^{r}\left( mc+n\right) ^{q}},$ $c\in\mathbb{N},$ in terms of certain zeta values.\textbf{ }Particular choices of $x$ and $y$ lead evaluation formulas for some Tornheim type $\sum_{m,n=1}^{\infty}\frac{1}{m^{p}% n^{r}\left( mc+n\right) ^{q}}$ and Euler type $\sum_{m,n=1}^{\infty}\frac {1}{n^{p}\left( mc+n\right) ^{q}}$ double series and their alternating analogues.