Under minimal assumptions on a function \(\psi\) we obtain wavelet-type frames of the form \[ \psi_{j,k}(x) = r^{(1/2)n j} \psi(r^j x - sk), \qquad j \in \mathbb{Z}, k \in \mathbb{Z}^n,\] for some \(r > 1\) and \(s > 0\). This collection is shown to be a frame for a scale of Triebel-Lizorkin spaces (which includes Lebesgue, Sobolev and Hardy spaces) and the reproducing formula converges in norm as well as pointwise a.e. The construction follows from a characterization of those operators which are bounded on a space of smooth molecules. This characterization also allows us to decompose a broad range of singular integral operators in terms of smooth molecules.