AbstractSuppose that you add rigid bars between points in the plane, and suppose that a constant fraction q of the points moves freely in the whole plane; the remaining fraction is constrained to move on fixed lines called sliders. When does a giant rigid cluster emerge? Under a genericity condition, the answer only depends on the graph formed by the points (vertices) and the bars (edges). We find for the random graph the threshold value of c for the appearance of a linear‐sized rigid component as a function of q, generalizing results of Kasiviswanathan, Moore, and Theran. We show that this appearance of a giant component undergoes a continuous transition for and a discontinuous transition for . In our proofs, we introduce a generalized notion of orientability interpolating between 1‐ and 2‐orientability, of cores interpolating between the 2‐core and the 3‐core, and of extended cores interpolating between the 2 + 1‐core and the 3 + 2‐core; we find the precise expressions for the respective thresholds and the sizes of the different cores above the threshold. In particular, this proves a conjecture of Kasiviswanathan, Moore, and Theran about the size of the 3 + 2‐core. We also derive some structural properties of rigidity with sliders (matroid and decomposition into components) which can be of independent interest.