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We present a photometric survey of the optical counterparts of ultraluminous X-ray sources (ULXs) observed with the Hubble Space Telescope in nearby (< 5 Mpc) galaxies. Of the 33 ULXs with Hubble & Chandra data, 9 have no visible counterpart, placing limits on their M_V of ~ -4 to -9, enabling us to rule out O-type companions in 4 cases. The refined positions of two ULXs place them in the nucleus of their host galaxy. They are removed from our sample. Of the 22 remaining ULXs, 13 have one possible optical counterpart, while multiple are visible within the error regions of other ULXs. By calculating the number of chance coincidences, we estimate that 13 +/- 5 are the true counterparts. We attempt to constrain the nature of the companions by fitting the SED and M_V to obtain candidate spectral types. We can rule out O-type companions in 20 cases, while we find that one ULX (NGC 253 ULX2) excludes all OB-type companions. Fitting with X-ray irradiated models provides constraints on the donor star mass and radius. For 7 ULXs, we are able to impose inclination-dependent upper and/or lower limits on the black holes mass, if the extinction to the assumed companion star is not larger than the Galactic column. These are NGC 55 ULX1, NGC 253 ULX1, NGC 253 ULX2, NGC 253 XMM6, Ho IX X-1, IC342 X-1 & NGC 5204 X-1. This suggests that 10 ULXs do not have O companions, while none of the 18 fitted rule out B-type companions., ApJS in press. 51 pages; 13 pages of text followed by colour figures and tables. Figure quality reduced to satisfy arXiv regulations, for high quality figure, contact j.c.gladstone@ualberta.ca or download accepted version from ApJS when available