  We consider nonlinear parabolic equations involving fractional diffusion of
the form $\partial_t u + (-\Delta)^s \Phi(u)= 0,$ with $0<s<1$, and solve an
open problem concerning the existence of solutions for very singular
nonlinearities $\Phi$ in power form, precisely $\Phi'(u)=c\,u^{-(n+1)}$ for
some $0< n<1$. We also include the logarithmic diffusion equation $\partial_t u
+ (-\Delta)^s \log(u)= 0$, which appears as the case $n=0$. We consider the
Cauchy problem with nonnegative and integrable data $u_0(x)$ in one space
dimension, since the same problem in higher dimensions admits no nontrivial
solutions according to recent results of the author and collaborators. The {\sl
limit solutions} we construct are unique, conserve mass, and are in fact
maximal solutions of the problem. We also construct self-similar solutions of
Barenblatt type, that are used as a cornerstone in the existence theory, and we
prove that they are asymptotic attractors (as $t\to\infty$) of the solutions
with general integrable data. A new comparison principle is introduced.
