Summary: In sleep research, applying finite mixture models to sleep characteristics captured through multiple data types, including self-reported sleep diary, a wrist monitor capturing movement (actigraphy), and brain waves (polysomnography), may suggest new phenotypes that reflect underlying disease mechanisms. However, a direct mixture model application is challenging because there are many sleep variables from which to choose, and sleep variables are often highly skewed even in homogenous samples. Moreover, previous sleep research findings indicate that some of the most clinically interesting solutions will be those that incorporate all three data types. Thus, we present two novel skewed variable selection algorithms based on the multivariate skew normal (MSN) distribution: one that selects the best set of variables ignoring data type and another that embraces the exploratory nature of clustering and suggests multiple statistically plausible sets of variables that each incorporate all data types. Through a simulation study, we empirically compare our approach with other asymmetric and normal dimension reduction strategies for clustering. Finally, we demonstrate our methods using a sample of older adults with and without insomnia. The proposed MSN-based variable selection algorithm appears to be suitable for both MSN and multivariate normal cluster distributions, especially with moderate to large-sample sizes.