We present results of a search for late-time radio emission and fast radio bursts (FRBs) from a sample of type-I superluminous supernovae (SLSNe-I). We used the Karl G. Jansky Very Large Array to observe 10 SLSN-I more than 5 yr old at a frequency of 3 GHz. We searched fast-sampled visibilities for FRBs and used the same data to perform a deep imaging search for late-time radio emission expected in models of magnetar-powered supernovae. No FRBs were found. One SLSN-I, PTF10hgi, is detected in deep imaging, corresponding to a luminosity of 1.2 × 1028 erg s−1. This luminosity, considered with the recent 6 GHz detection of PTF10hgi in Eftekhari et al., supports the interpretation that it is powered by a young, fast-spinning (∼ms spin period) magnetar with ∼15 M of partially ionized ejecta. Broadly, our observations are most consistent with SLSNe-I being powered by neutron stars with fast spin periods, although most require more free-free absorption than is inferred for PTF10hgi. We predict that radio observations at higher frequencies or in the near future will detect these systems and begin constraining properties of the young pulsars and their birth environments.