  We consider self-similar solutions with finite mass to Smoluchowski's
coagulation equation for rate kernels that have homogeneity zero but are
possibly singular such as Smoluchowski's original kernel. We prove pointwise
exponential decay of these solutions under rather mild assumptions on the
kernel. If the support of the kernel is sufficiently large around the diagonal
we also proof that $\lim_{x \to \infty} \frac{1}{x} \log
\Big(\frac{1}{f(x)}\Big)$ exists. In addition we prove properties of the
prefactor if the kernel is uniformly bounded below.
