It depends on a lot of things.
Most sites I do not check my links often. Especially sites I build PBN links to.
If I need to check my own links, I have used Ahrefs, Majestic.
The only sites I check links on are sites which I am not building links to. If I do build links they are live because I check my link sources (PBN, Web 2.0 and Guest Post).
So in short, if I need to check links on a site I am not building links to the site, so I go with webmaster tools. I know of all my 'live' links and if my rankings drop for some reason, I will grab a full link report with Ahrefs and see if I notice any reason for the drop. But this is not a common situation.