We consider the problem of $q$-colouring a $k$-uniform random hypergraph, where $q,k \geq 3$, and determine the rigidity threshold. For edge densities above the rigidity threshold, we show that almost all solutions have a linear number of vertices that are linearly frozen, meaning that they cannot be recoloured by a sequence of colourings that each change the colour of a sublinear number of vertices. When the edge density is below the threshold, we prove that all but a vanishing proportion of the vertices can be recoloured by a sequence of colourings that recolour only one vertex at a time. This change in the geometry of the solution space has been hypothesised to be the cause of the algorithmic barrier faced by naive colouring algorithms. Our calculations verify predictions made by statistical physicists using the non-rigorous cavity method. The traditional model for problems of this type is the random colouring model, where a random hypergraph is chosen and then a random colouring of that hypergraph is selected. However, it is often easier to work with the planted model, where a random colouring is selected first, and then edges are randomly chosen which respect the colouring. As part of our analysis, we show that up to the condensation phase transition, the random colouring model is contiguous with respect to the planted model. This result is of independent interest.