We study push-pull rumour spreading in ultra-small-world models for social networks where the degrees follow a power-law distribution. In a non-geometric setting, Fountoulakis, Panagiotou and Sauerwald have shown that rumours always spread ultra-fast (SODA 2012), i.e. in doubly logarithmic time. On the other hand, Janssen and Mehrabian have found that rumours spread slowly (polynomial time) in a spatial preferential attachment model (SIDMA 2017). We study the question systematically for the model of Geometric Inhomogeneous Random Graphs (GIRGs). Our results are two-fold: first, with Euclidean geometry slow, fast (polylogarithmic) and ultra-fast rumour spreading may occur, depending on the exponent of the power law and the strength of the geometry in the networks, and we fully characterise the phase boundaries in between. The regimes do not coincide with the graph distance regimes, i.e., polylogarithmic or even polynomial rumour spreading may occur even if graph distances are doubly logarithmic. We expect these results to hold with little effort for related models, e.g. Scale-Free Percolation. Second, we show that rumour spreading is always (at least) fast in a non-metric geometry. The considered non-metric geometry allows to model social connections where resemblance of vertices in a single attribute, such as familial kinship, already strongly indicates the presence of an edge. Euclidean geometry fails to capture such ties. For some regimes in the Euclidean setting, the efficient pathways for spreading rumours differ from previously identified paths. For example, a vertex of degree $d$ can transmit the rumour to a vertex of larger degree by a chain of length $3$, where one of the two intermediaries has constant degree, and the other has degree $d^{c}$ for some constant $c<1$. Similar but longer chains of vertices, all having non-constant degree, turn out to be useful as well.