It is widely recognized that missing entries can render some taxa unstable in phylogenetic parsimony analyses, leading to multiple most parsimonious trees and poorly resolved consensus trees that fail to reveal strictly supported relationships (relationships that are common to all the most parsimonious trees). An illustrated and non-technical review of strategies for ameliorating this problem, including safe taxonomic reduction and the strict reduced consensus method, is presented. Importantly, taxon instability is not necessarily a problem of missing entries per se, but it may be exacerbated by missing entries. A less widely recognized, but nonetheless important potential impact of missing entries and instability is obfuscation of the levels of the support provided by the data for particular relationships. Low bootstrap proportions and/or decay indices may reflect the instability of a subset of the taxa rather than more systemic deficiencies of the data. Even when there is no gross instability, relative instability can negatively impact upon measures of support. Rather than eschewing measures of support because of such problems, phylogeneticists should use methods of evaluating support that are not unduly affected by relatively unstable taxa. The majority-rule reduced consensus method provides one method that can be used to enhance bootstrap and jackknife analyses so as to reveal ‘hidden support’. Double decay analysis provides an analogous extension to traditional decay analysis. Non-technical summaries of these methods are presented and their potential benefits and limitations are discussed and illustrated.