The design of the CMS detector is specially optimized for muon measurements and includes gas-ionization detector technologies to make up the muon system. Cathode strip chambers (CSC) with both tracking and triggering capabilities are installed in the forward region. The first stage of muon reconstruction deals with information from within individual muon chambers and is thus called local reconstruction. First, rechits with well-defined locations are build from the electrical signals induced by the charged particles traversing the chamber. Next, from the reconstructed hits, straight-line track segments are build within each chamber. The local reconstruction becomes particularly challenging at high instantaneous luminosities such as the expected at HL-LHC. The high rate of particles traversing the detectors leads to increased rate of spurious rechits and segments thus increasing the combinatorial backgrounds. In this respect, work on improving the current and developing new algorithms is essential and is in progress. This talk presents the local reconstruction algorithms used in the CMS cathode strip chambers. Their performance as well as ongoing efforts towards HL-LHC will be discussed.