An impact oscillator is a periodically driven system that hits a wall when its amplitude exceeds a critical value. We study impact oscillations where collisions with the wall are with near-zero velocity (grazing impacts). A characteristic feature of grazing impact dynamics is a geometrically converging series of transitions from a nonimpacting period-1 orbit to period-M orbits that impact once per period with M=1,2,ellipsis. In an experiment we explore the dynamics in the vicinity of these period-adding transitions. The experiment is a mechanical impact oscillator with a precisely controlled driving strength. Although the excitation of many high-order harmonics in the experiment appeared unavoidable, we characterize it with only three parameters. Despite the simplicity of this description, good agreement with numerical simulations of an impacting harmonic oscillator was found. Grazing impact dynamics can be described by mappings that have a square-root singularity. We evaluate several mappings, both for instantaneous impacts and for impacts that involve soft collisions with a yielding wall. As the square-root singularity appears persistent in the reduction of the dynamics to mappings, and because impact dynamics appears insensitive to experimental nonidealities, the characteristic bifurcation scenario should be observed in a wide class of experimental systems.