Compressive Sensing Through MATLAB Codes
L1magic. This is the toolbox associated with the original paper.
CompSens. This looks like it's in C, but you could possibly call it with mex -- not sure.
Model-based compressive sensing toolbox.



  
  Most of the code is plain Matlab code
  Each folder in the package consists of a CS recovery algorithm based on a particular signal model, and a script that tests that recovery algorithm. The names of the scripts typically end with '_example.m'
  



Manopt. Possibly what you use to actually make the algorithms included in other toolboxes.



  Optimization on manifolds is a powerful paradigm to address nonlinear optimization problems.



Some examples from UMich.
Sparsity. 



  This toolbox implements several algorithms to compute sparse expansion in redundant dictionaries and to solve inverse problems with sparse regularization (and also TV regularization).


But all of that, and more, is included in this list of toolboxes.

I've found the hard part is finding psuedocode -- that's where they actually describe the algorithm. Here are some examples of algorithms that included the psuedocode: 


one paper that includes the actual algorithm (and the actual code available here in the function IST.
a beginners guide to compressed sensing.
Orthogonal matching pursuit
CS-KLT
Some more algorithms from Microsoft
MCMC for universal CS
An example
Compressive sensing of streams of pulses
GPU Accelerated Greedy Algorithms for Compressed Sensing
Adaptive sensing camera
Compressive sensing meets game theory