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The morphological properties of high-redshift galaxies are investigated using a sample of 507 objects (I < 22.0 mag) from the Hubble Space Telescope (HST) Medium Deep Survey. Independent visual morphological classifications for each galaxy are used to quantify the statistical uncertainties in the galaxy classifications. Visual classifications are found to agree well for I < 21 mag. Fainter than I = 21 mag significant disagreements are seen in the independent visual classifications of late-type systems with T > 7, merging systems, and peculiar galaxies. The classifications of these systems are shown to be some- what subjective. Objective classifications based upon measurements of central concentration and asymmetry for the Medium Deep Survey sample are presented. These classifications are calibrated using measurements of structural parameters for an artificially redshifted sample of local objects. Morphologically segregated number counts using both sets of visual classifications and objective classifications support the conclusion that the observed galaxy counts agree with no-evolution predictions for the elliptical and spiral populations, as reported in Glazebrook et al. (1995a). A major conclusion is that the large overdensity of merging/peculiar/irregular galaxies relative to the predictions of no-evolution models (reported by Glazebrook et al. 1995a) is confirmed. However, the shape of the faint-end (I > 21.0 mag) number count relation for peculiar objects is sensitive to the large systematic uncertainties inherent in the visual classification of these objects. Despite this caveat, the frequency of objects showing clear evidence for tidal interactions (e.g., tidal tails) in the HST sample is at least 50% larger than it is among nearby galaxies, at the 2 σ level. Relatively few "chain galaxies" are seen among the sample of peculiar objects, suggesting that these systems do not form a large component of the peculiar galaxy population at I < 22 mag.