The striking diversity of bird beak shapes is an outcome of natural selection, yet the relative importance of the limitations imposed by the process of beak development on generating such variation is unclear. Untangling these factors requires mapping developmental mechanisms over a phylogeny far exceeding model systems studied thus far. We address this issue with a comparative morphometric analysis of beak shape in a diverse group of songbirds. Here we show that the dynamics of the proliferative growth zone must follow restrictive rules to explain the observed variation, with beak diversity constrained to a three parameter family of shapes, parameterized by length, depth and the degree of shear. We experimentally verify these predictions by analysing cell proliferation in the developing embryonic beaks of the zebra finch. Our findings indicate that beak shape variability in many songbirds is strongly constrained by shared properties of the developmental programme controlling the growth zone.