The J-1 Visitor Exchange Program visa is generally the preferred choice for non-US citizens and the visa is organised and sponsored by the sponsoring agency (usually the Flight School). This may be issued for 18 mo with a 6 mo extension.
The J-1 allows the applicant to gain "related [paid] practical experience" after completing training and for the remainder of the visa.
Last i checked though, the J-1 may not be issued if the applicant already holds a certificate/rating above the PPL/IR level (ie not a professional license). Check carefully with individual flight schools and ask for the reference they base their answer on.
The M-1 visa, as cfimei correctly stated, allows for 1 month for 4 moths training of "practical expereince". Realistically, this is a very limited option to gain paid experience. Preference would be for the J-1.
If you have the interest and money to study at an accredited post-secondary establishement, there is the F-1 visa which as some part-time "practical [paid] expereince" opportunites, but read the small print re. how many hours you may "work"
J-1 visa requires an US Dept. of Education "accredited" program (nothing to do with the FAA) AND be a Part 141 Approved School (everything to do with the FAA), so there are fewer of them around.
US ICE (immigration and Customs Enforcement) are looking carefully at US Schools right now. There is a background check required of the student (usual criminal type check) costing US$130. See AOPA for more.
It is still a worthwhile endevour if planned carefully....ask around for previous students at the schools youre looking at.
Hope this helps.
ENG Brit