Transmigrating cells often need to deform cell body and nucleus to pass through micrometer-sized pores in extracellular matrix scaffolds. Furthermore, chemoattractive signals typically guide transmigration, but the precise interplay between mechanical constraints and signaling mechanisms during 3D matrix invasion is incompletely understood and may differ between cell types. Here, we used Direct Laser Writing to fabricate 3D cell culture scaffolds with adjustable pore sizes (2–10 μm) on a microporous carrier membrane for applying diffusible chemical gradients. Mouse embryonic fibroblasts invade 10 μm pore scaffolds even in absence of chemoattractant, but invasion is significantly enhanced by knockout of lamin A/C, a known regulator of cell nucleus stiffness. Nuclear stiffness thus constitutes a major obstacle to matrix invasion for fibroblasts, but chemotaxis signals are not essential. In contrast, epithelial A549 cells do not enter 10 μm pores even when lamin A/C levels are reduced, but readily enter scaffolds with pores down to 7 μm in presence of chemoattractant (serum). Nuclear stiffness is therefore not a prime regulator of matrix invasion in epithelial cells, which instead require chemoattractive signals. Microstructured scaffolds with adjustable pore size and diffusible chemical gradients are thus a valuable tool to dissect cell-type specific mechanical and signaling aspects during matrix invasion.