The most common proposals are directed to the Census Bureau. Depending on the restricted data necessary, the Agency for Healthcare and Research Quality (AHRQ), the Bureau of Economic Analysis (BEA), the Bureau of Labor Statistics (BLS), or the National Center for Health Statistics (NCHS) may also need to review and approve the proposal.  Each has different application processes and guidelines.  These links lay out the essential information: 

All initial proposal applications need to be submitted through the Application Portal for Restricted Data for Federal Statistics.

Other Helpful Resources:

FSRDC Server Software        
Gauss Stata Matlab & toolboxes PBS Pro Intel Composer XE
NX Enterprise R SAS SAS (Dataflux) SUDAAN
GEODa Tomlab Knittro Madd QGIS
StatTransfer Python - Anaconda Fortran Perl Tex/LaTex