I'm currently using NeuN for soma detection and I'm looking for a good axon staining: other of my group used B-tubulin III with poor results. I'll try to use also the MAP-2 and maybe to improve the B-tubulin III stain protocol.
For soma and generic dendrite staining, I suggest you may use Neu-N and Tuj1 or MAP2 respectively.
However, If you would like to specifically characterize the axonal extensions, you may try anti Tau-1 antibody which has high clonal specificity to growing axons. Ofcourse B-tubulin III(i.e Tuj1) can stain the axons, but with lot many other dendrites which may seem bit messy in high density cultures.
Hence you may select species reactivity of Anti-NeuN and Anti-Tau1 and use them in conjunction for Soma + Axon staining.
MAP2 is usually a better marker for dendrites, neurofilament is a good marker for axons (or SMI-32, the unphosphorylated form is good for outlining axons in the grey matter). We use a combination of Neun, MAP2 and neurofilament (chicken one from millipore is very good) that together will label the vast majority of the cell.