AASHTO LRFD is applicable, usually supplemented by proeject specific design criteria. We typically incorporate some of PTI's recommendations for stay cable design, testing, and installation.
Also, ASCE has a slim booklet "Guidelines for the Design of Cable Stayed Bridges".
THe AASHTO Specs are applicable. The local (State DOT?) construction specs may also apply. There is also a Guide Spec for segmental concrete construction that may be helpful if it is made of concrete. Good Luck.