We demonstrate the advantages of using a single-crystal sapphire fiber to perform Raman spectroscopy over using a standard glass fiber. The narrow Raman peaks and the low background signal of a single-crystal sapphire fiber allow for the detection of weaker Raman signals or Raman signals much closer to the incident laser wavelength than with a standard glass fiber, which has a strong broadband Raman spectrum of its own. The Raman spectra of calcite, single-walled carbon nanotubes, and aqueous sodium carbonate solution were collected using a $50\mathrm{cm}$ long, $200\mathrm{\mu m}$ diameter sapphire fiber, and then they were compared with the results using a $50\mathrm{cm}$ long, $100\mathrm{\mu m}$ diameter glass fiber. The Raman spectrum of each tested material becomes indistinguishable due to the background Raman signal of the glass fiber but can easily be seen using the sapphire fiber, which has only a few narrow Raman peaks of its own.