Linear growth functionals (such as the minimal surface functional) are usually minimised over the space BV of functions of bounded variation. However, if we pass to the vectorial case and replace the full by the symmetric gradient, then Ornstein's Non-Inequality rules out coerciveness of linear growth functionals on BV. We are thereby lead to examine generalised minima over the space BD of functions of bounded deformation: For such maps the symmetric distributional gradients are finite Radon measures. In this talk I aim to give a comprehensive regularity analysis for generalised minimisers both in the convex and the symmetric-semiconvex framework. The results presented therein are partially obtained in collaboration with J. Kristensen (Oxford).