Background: Neutrinoless double-β decay, if observed, would reveal physics beyond the standard model of particle physics; namely, it would prove that neutrinos are Majorana fermions and that the lepton number is not conserved. Purpose: The analysis of the results of neutrinoless double-β decay observations requires an accurate knowledge of several nuclear matrix elements (NME) for different mechanisms that may contribute to the decay. We provide a complete analysis of these NME for the decay of the ground state (g.s.) of 48Ca to the g.s. 01+ and first excited 02+ state of 48Ti. Method: For the analysis we used the nuclear shell model with effective two-body interactions that were fine-tuned to describe the low-energy spectroscopy of pf-shell nuclei. We checked our model by calculating the two-neutrino transition probability to the g.s. of 48Ti. We also make predictions for the transition to the first excited 02+ state of 48Ti. Results: We present results for all NME relevant for the neutrinoless transitions to the 01+ and 02+ states, and using the lower experimental limit for the g.s. to g.s. half-life, we extract upper limits for the neutrino physics parameters. Conclusions: We provide accurate NME for the two-neutrino and neutrinoless double-β decay transitions in the A=48 system, which can be further used to analyze the experimental results of double-β decay experiments when they become available.