This study explores the space density and properties of active galaxies to
z similar or equal to 0.8. We have investigated the frequency and nature of
unresolved nuclei in galaxies at moderate redshift as indicators of nuclea
r activity such as active galactic nuclei (AGNs) or starbursts. Candidates
are selected by fitting imaged galaxies with multicomponent models using ma
ximum likelihood estimate techniques to determine the best model fit. We se
lect those galaxies requiring an unresolved, point-source component in the
galaxy nucleus, in addition to a disk and/or bulge component, to adequately
model the galaxy light. We have searched 70 WFPC2 images primarily from th
e Medium Deep Survey for galaxies containing compact nuclei. In our survey
of 1033 galaxies, the fraction containing an unresolved nuclear component g
reater than or equal to 3% of the total galaxy light is 16% +/- 3% correcte
d for incompleteness and 9% +/- 1% for nuclei greater than or equal to 5% o
f the galaxy light. Spectroscopic redshifts have been obtained for 35 of ou
r AGN/starburst candidates, and photometric redshifts are estimated to an a
ccuracy of sigma(z) similar or equal to 0.1 for the remaining sample. In th
is paper, the first of two for this study, we present the selected Hubble S
pace Telescope imaged galaxies having unresolved nuclei and discuss the sel
ection procedure. We also present the ground-based spectroscopy for these g
alaxies as well as the photometric redshift predictions for those galaxies
without spectra.