Extreme events arising in georeferenced processes can take various forms, such as occurring in isolated patches or stretching contiguously over large areas, and can further vary with the spatial location and the extremeness of the events. We use excursion sets above threshold exceedances in data observed over a two-dimensional grid of rectangular pixels to propose a general family of coefficients that assess spatial-extent properties relevant for risk assessment, and study five candidate coefficients from this family. These coefficients are defined locally and interpreted as a spatial distance from a reference site where the threshold is exceeded. We develop statistical inference and discuss robustness to boundary effects and resolution of the pixel grid. To statistically extrapolate coefficients towards very high threshold levels, we formulate a semiparametric model and estimate a parameter characterizing how coefficients scale with the quantile level of the threshold. The utility of the new coefficients is illustrated through simulated data, as well as in an application to gridded daily temperature in continental France. We find notable differences in estimated coefficient maps between climate model simulations and observation-based reanalysis.