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We prove the existence of nonseparable, orthonormal, compactly supported wavelet bases for $L^2\left({ℝ}^2\right)$ of arbitrarily high regularity by using some basic techniques of algebraic and differential geometry. We even obtain a much stronger result: “most” of the orthonormal compactly supported wavelet bases for $L^2\left({ℝ}^2\right)$, of any regularity, are nonseparable