We identify a large number of galaxy clusters in the Lick map using an algorithm based on an overdensity criterion. The resulting catalogues contain ~ 6000 clusters out of which 753 are Abell clusters. We determine ellipticities and major axis orientations for a suitable subset of this sample, including 397 Abell clusters. We find that the distribution of projected axial ratios is approximately Gaussian with a mean of ~ 0.6 and a standard deviation of ~ 0.2. We investigate methods to invert the distribution of apparent axial ratios in order to obtain the distribution of intrinsic axial ratios. We find that the apparent shapes of clusters in our sample are inconsistent with those expected from a population of oblate spheroids seen in projection. The observed distribution could come from a population of prolate spheroids with a Gaussian distribution of axial ratios with mean ~ 0.5 and standard deviation ~ 0.15. However, we show that in the general case where objects are triaxial or there is a mixture of oblate and prolate configurations, the inversion to intrinsic shapes is not necessarily unique. There appears to be a significant increase of ellipticity with redshift out to the limit of our sample, z ~ 0.15. This could be due to a distance dependent contamination by projection effects or to intrinsic evolution in the cluster population.