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(abridged) Deep multi-epoch Sloan Digital Sky Survey data in a 275 squaredegrees area along the celestial equator (SDSS stripe 82 = S82) allowed us tosearch for extremely faint ($i>21$) objects with proper motions larger than0.14 arcsec/yr. We classify 38 newly detected objects with low-resolutionoptical spectroscopy using FORS1 @ ESO VLT. All 22 previously known L dwarfs inS82 have been detected in our high proper motion survey. However, 11 of theknown L dwarfs have smaller proper motions (0.01$<$$\mu$$<$0.14 arcsec/yr).Although S82 was already one of the best investigated sky regions with respectto L and T dwarfs, we are able to classify 13 new L dwarfs. We have also foundeight new M7.5-M9.5 dwarfs. Four new cool white dwarfs (CWDs) discovered by usare about 1-2 mag fainter than those previously detected in SDSS data. All newL-type, late-M and CWD objects show thick disk and halo kinematics. There are13 objects, mostly with uncertain proper motions, which we initially classifiedas mid-M dwarfs. Among them we have found 9 with an alternative subdwarfclassification (sdM7 or earlier types), whereas we have not found any newspectra resembling the known ultracool ($>$sdM7) subdwarfs. Some M subdwarfcandidates have been classified based on spectral indices with largeuncertainties. We failed to detect new nearby ($d<50$ pc) L dwarfs, probablybecause the S82 area was already well-investigated before. With our survey wehave demonstrated a higher efficiency in finding Galactic halo CWDs thanprevious searches. The space density of halo CWDs is according to our resultsabout 1.5-3.0 $\times$ 10$^{-5}$ pc$^{-3}$.

Extremely faint high proper motion objects from SDSS stripe 82 - Optical classification spectroscopy of about 40 new objects