We study the behavior of Haar coefficients in Besov and Triebel–Lizorkin spaces on {mathbb R}, for a parameter range in which the Haar system is not an unconditional basis. First, we obtain a range of parameters, extending up to smoothness s<1, in which the spaces F^s_{p,q} and B^s_{p,q} are characterized in terms of doubly oversampled Haar coefficients (Haar frames). Secondly, in the case that 1/p<s<1 and fin B^s_{p,q}, we actually prove that the usual Haar coefficient norm, Vert {2^jlangle f, h_{j,mu }rangle }_{j,mu }Vert _{b^s_{p,q}} remains equivalent to Vert fVert _{B^s_{p,q}}, i.e., the classical Besov space is a closed subset of its dyadic counterpart. At the endpoint case s=1 and q=infty , we show that such an expression gives an equivalent norm for the Sobolev space W^{1}_p({mathbb R}), 1<p<infty , which is related to a classical result by Bočkarev. Finally, in several endpoint cases we give optimal inclusions between B^s_{p,q}, F^s_{p,q}, and their dyadic counterparts.