  We experimentally investigate the stability of a quantum gas with repulsive
interactions in an optical 1D lattice subjected to periodic driving.
Excitations of the gas in the lowest lattice band are analyzed across the
complete stability diagram, from slow to fast driving frequencies and from weak
to strong driving strengths. To interpret our results, we expand the
established analysis based on parametric instabilities to include modulational
instabilities. Extending the concept of modulational instabilities from static
to periodically driven systems provides a convenient mapping of the stability
in a static system to the cases of slow and fast driving. At intermediate
driving frequencies, we observe an interesting competition between modulational
and parametric instabilities. We experimentally confirm the existence of both
types of instabilities in driven systems and probe their properties. Our
results allow us to predict stable and unstable parameter regions for the
minimization of heating in future applications of Floquet driving.
