We present spectra of 36 nearby star candidates and 3 red giant candidates, identified in the DENIS database. 32 of the dwarf candidates are nearby red dwarfs, with spectral types from M 5.5 to M 8.5. Out of 11 targets with low proper motion (μ < 0.1 arcsec yr^-1) but a Reduced Proper Motion above an inclusive threshold, 9 are red dwarfs. The 4 contaminants are all reddened F-K main sequence stars, and could have been eliminated by checking for small well-known high latitude molecular clouds. These stars might be of interest as probes of interstellar absorption. For the red dwarfs we derive spectral types and spectroscopic distances, using a new calibration of the PC3 spectral index to absolute magnitudes in the I, J, H and K photometric bands. We confirm 2 new members of the 12 pc volume (2 new M 8.5), and one M 7.5 NLTT object closer than 10 pc; and show that one quarter of the stars with photometric distances under 30 pc have too small a proper motion for inclusion in the NLTT catalog.