We investigate the pulsar population in the Galactic center and discuss novel observational strategies to discover pulsars in orbit around the black hole Sgr A*. In order to motivate these searches, we first use published multi-wavelength observations to set constraints on the allowed population of radio pulsars in the Galactic center. Since the parameters of any search depend strongly on the scattering of pulsar signals in the interstellar medium, we study the distribution of free electrons along the line of sight to the Galactic center. Next, we conduct a single pulse analysis on pulses from the radio-emitting magnetar J1745-2900. This analysis provides additional information on the amount of scattering along the line of sight and presents an opportunity to study a rare type of neutron star whose presence in the vicinity of Sgr A* may have important implications for the pulsar population in the Galactic center. Using the Karl~G.~Jansky Very Large Array (VLA) we conduct two Galactic center pulsar searches using newly developed pulsar observing modes. Finally, we use the new fast-visibility VLA observing mode to provide the first arcsecond localization of a Fast Radio Burst.