Many processes in eukaryotic cells, including the crawling motion of the whole cell, rely on the growth of branched
actin networks from surfaces. In addition to their well-known role in generating propulsive forces, actin networks can also sustain
substantial pulling loads thanks to their persistent attachment to the surface from which they grow. The simultaneous network
elongation and surface attachment inevitably generate a force that opposes network growth. Here, we study the local dynamics
of a growing actin network, accounting for simultaneous network elongation and surface attachment, and show that there exist
several dynamical regimes that depend on both network elasticity and the kinetic parameters of actin polymerization. We characterize this in terms of a phase diagram and provide a connection between mesoscopic theories and the microscopic dynamics
of an actin network at a surface. Our framework predicts the onset of instabilities that lead to the local detachment of the network
and translate to oscillatory behavior and waves, as observed in many cellular phenomena and in vitro systems involving actin
network growth, such as the saltatory dynamics of actin-propelled oil drops.