  The advent of the Hubble Space Telescope (HST) has provided images of
galaxies at moderate and high redshifts and changed the scope of galaxy
morphologies considerably. It is evident that the Hubble Sequence requires
modifications in order to incorporate all the various morphologies one
encounters at such redshifts. We investigate and compare different approaches
to quantifying peculiar galaxy morphologies on images obtained from the Medium
Deep Survey (MDS) and other surveys using the Wide Field Planetary Camera 2
(WFPC2) on board the HST, in the I band (filter F814W). We define criteria for
peculiarity and put them to use on a sample of 978 galaxies, classifying them
by eye as either normal or peculiar. Based on our criteria and on concepts
borrowed from digital image processing we design a set of four purely
morphological parameters, which comprise the overall texture (or
``blobbiness'') of the image; the distortion of isophotes; the filling-factor
of isophotes; and the skeleta of detected structures. We also examine the
parameters suggested by Abraham et al. (1995). An artificial neural network
(ANN) is trained to distinguish between normal and peculiar galaxies. While the
majority of peculiar galaxies are disk-dominated, we also find evidence for a
significant population of bulge-dominated peculiars. Consequently, peculiar
galaxies do not all form a ``natural'' continuation of the Hubble sequence
beyond the late spirals and the irregulars. The trained neural network is
applied to a second, larger sample of 1999 WFPC2 images and its probabilistic
capabilities are used to estimate the frequency of peculiar galaxies at
moderate redshifts as $35 \pm 15 %$.
