I know little. A Search under those names would turn up better results, and there is a steel FAQ here.
But, VG-10 is a European steel (Sweedish?) that is stainless and takes an aggressive edge. ATS-34 is a Japanese version of 154-CM. Crucible first came out with 154-CM in the 70s and it was sailed as the grail of stainless steel. I think it then became hard to get in knvie making sizes and/or the steel quality went down. A Japanese firm (I forget the one) made ATS and made a killing on it. It was cheap and better than the 154-CM. Crucible's new offering of 154-CM, as far as I know, is once again high quality, though the only difference between ATS34 and 154CM is where it is made, in essence.
BG-42 is a ball bearing steel, again a European steel. it is VERY clean, as it goes through a VIM-VAR process: Vacuum Induction Melt-Vacuum somethign or other Re-melt, I think. Regardless, it is a very clean steel. I think it pretty much is ATS with vanadium in it. Vanadium carbides are the secret to the latest super alloys.
I hope this helps in some way.