We describe longitudinally pumped solid-state lasers from both experimental and numerical points of view. We explore in detail the particular behaviors of non-degenerate as well as partially or fully degenerate cavities. Our work provides some insight on the formation of high-order Ince-Gauss or Laguerre-Gauss modes. Namely we show analitically, numerically, and experimentally that it is impossible to excite IGnm modes (n≠m) with a Gaussian pump in non-degenerate cavities, contrary to a common belief. We find the emergence of quasi-Ince and quasi-Laquerre Gaussian modes in the vicinity of degeneracy points, in full agreement with observations. Numerical simulations are based on the Fox-Li algorithm. Experiments are conducted with diode-pumped Nd-doped lasers. We discuss extensions of this work towards the generation of original transverse modes.