These solvers do not fit into other categories, or else they are waiting to be categorized.
The simplest algorithm is iterative hard-thresholding (IHT) which directly tries to solve the best subset selection problem. It has been analyzed and can provide some recovery guarantees. Thomas Blumensath has a collection of greedy and hard thresholding algorithms (GreedLab and HardLab).
Smoothing l0 quasi-norm
A coordinate descent algorithm for l1 functions is proposed in 
- sparseMRI by Michael Lustig and Stanford co-authors. For MRI.
- l1-spirit by Michael Lustig. This is custom for MRI reconstruction, and is very well-done code (MATLAB, which calls mex files and can call CUDA executables to take advantage of a fast graphics card).
- KF-CS: Kalman Filtered CS (and other sequential CS algorithms)
- k-t FOCUSS. FOCUSS is a type of iteratively-reweighted least-squares (IRLS).
- Cycling matching pursuit (CMP)] with code. They say it's similar to OMP