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In the article we study the Neumann $(p,q)$-eigenvalue problems in boundedH\"older $\gamma$-singular domains $\Omega_{\gamma}\subset \mathbb{R}^n$. Inthe case $1<p<\infty$ and $1<q<p^{*}_{\gamma}$ we prove solvability of thiseigenvalue problem and existence of the minimizer of the associated variationalproblem. In addition, we establish some regularity results of theeigenfunctions and some estimates of $(p,q)$-eigenvalues. \

On the Neumann $(p,q)$-eigenvalue problem in H\"older singular domains