I don't know about old ACE programs, but AFAIK recent ones use .ace, .c00, .c01, .c02 etc.
I'm not aware of any archive format that "natively" uses the .001 convention - my guess is that this was invented by warez groups to counteract the confusion that can occur with the normal RAR naming conventions: .rar is actually the first file in the archive, but it's listed last alphabetically, file .rxx is actually the xx+2th file, a release with the last file being .rxx actually has xx+2 files, etc.