At zero temperature a disordered solid corresponds to a local minimum in the energy landscape. As the temperature is raised or the system is driven with a mechanical load, the system explores different minima via dynamical events in which particles rearrange their relative positions. We have shown recently that the dynamics of particle rearrangements are strongly correlated with a structural quantity associated with each particle, "softness", which we can identify using supervised machine learning. Particles of a given softness have a well-defined energy scale that governs local rearrangements; because of this property, softness greatly simplifies our understanding of glassy dynamics. Here we investigate the correlation of softness with other commonly used structural quantities, such as coordination number and local potential energy. We show that although softness strongly correlates with these properties, its predictive power for rearrangement dynamics is much higher. We introduce a useful metric for quantifying the quality of structural quantities as predictors of dynamics. We hope that, in the future, authors introducing new structural measures of dynamics will compare their proposals quantitatively to softness using this metric. We also show how softness correlations give insight into rearrangements. Finally, we explore the physical meaning of softness using unsupervised dimensionality reduction and reduced curve-fitting, models, and show that softness can be recast in a form that is amenable to analytical treatment.