The ability to survey polymorphism on a genomic scale has enabled genome-wide scans for the targets of natural selection. Theory that connects patterns of genetic variation to evidence of natural selection most often assumes a diallelic locus and no recurrent mutation. Although these assumptions are suitable to selection that targets single nucleotide variants, fundamentally different types of mutation generate abundant polymorphism in genomes. Moreover, recent empirical results suggest that mutationally complex, multiallelic loci including microsatellites and copy number variants are sometimes targeted by natural selection. Given their abundance, the lack of inference methods tailored to the mutational peculiarities of these types of loci represents a notable gap in our ability to interrogate genomes for signatures of natural selection. Previous theoretical investigations of mutation-selection balance at multiallelic loci include assumptions that limit their application to inference from empirical data. Focusing on microsatellites, we assess the dynamics and population-level consequences of selection targeting mutationally complex variants. We develop general models of a multiallelic fitness surface, a realistic model of microsatellite mutation, and an efficient simulation algorithm. Using these tools, we explore mutation-selection-drift equilibrium at microsatellites and investigate the mutational history and selective regime of the microsatellite that causes Friedreich's ataxia. We characterize microsatellite selective events by their duration and cost, note similarities to sweeps from standing point variation, and conclude that it is premature to label microsatellites as ubiquitous agents of efficient adaptive change. Together, our models and simulation algorithm provide a powerful framework for statistical inference, which can be used to test the neutrality of microsatellites and other multiallelic variants.