  We consider a class of reaction-diffusion equations of Fisher-KPP type in
which the nonlinearity (reaction term) $f$ is merely $C^1$ at $u=0$ due to a
logarithmic competition term. We first derive the asymptotic behavior of
(minimal speed) traveling wave solutions that is, we obtain precise estimates
on the decay to zero of the traveling wave profile at infinity. We then use
this to characterize the Bramson shift between the traveling wave solutions and
solutions of the Cauchy problem with localized initial data. We find a phase
transition depending on how singular $f$ is near $u=0$ with quite different
behavior for more singular $f$. This is in contrast to the smooth case, that
is, when $f \in C^{1,\delta}$, where these behaviors are completely determined
by $f'(0)$. In the singular case, several scales appear and require new
techniques to understand.
