A fundamental problem in partial differential equations is to understand the size and geometric structure of removable sets for solutions to a given PDE. Geometric Measure Theory provides the necessary machinery in order to quantify and geometrically describe such removable sets. In the Euclidean case classical results have been obtained by various authors including Carleson, David, Mattila, Harvey and Polking. This talk will focus on recent work which extends several of these results in the setting of Carnot groups. I will also discuss how new singular integrals emerge naturally in our proofs, motivating several open questions in geometric harmonic analysis.