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We show that, for suitable enumerations, the multivariate Haar system is aSchauder basis in the classical Sobolev spaces on $\mathbb R^d$ withintegrability $1<p<\infty$ and smoothness $1/p-1<s<1/p$. This complementsearlier work by the last two authors on the unconditionality of the Haar systemand implies that it is a {conditional} Schauder basis for a nonempty opensubset of the $(1/p,s)$-diagram. The results extend to (quasi-)Banach spaces ofHardy-Sobolev and Triebel-Lizorkin type in the range of parameters$\frac{d}{d+1}<p<\infty$ and $\max\{d(1/p-1),1/p-1\}<s<\min\{1,1/p\}$, which isoptimal except perhaps at the end-points.

The Haar system as a Schauder basis in spaces of Hardy-Sobolev type