Interesting question, especially because this election cycle has been so different.
On the Democratic side, I imagine that Clinton already has a vetting process going on quietly (and unless she is facing Kasich, my bet is her choice is someone like Sen. Kaine). I am guessing that Sanders has not even gotten to thinking about it. If he does get the opportunity, he may have to agree to a "brokered" establishment VP candidate.
On the GOP side, with the contested convention, all bets are off. The VP candidate may well be forced onto the ticket, or a convention deal between a couple of candidates may determine the VP slot.
If the GOP candidates have a choice,Trump will definitely want someone politically experienced - as Silly American said, the ideal "fantasy candidate" would be a woman Governor (as long as it is not Palin
). Similarly Cruz would need an experienced person who would alleviate mainstream concerns (someone like Scott Walker, but with some charisma). If he has the choice, Kasich has the "luxury" to get someone young and telegenic (Gov. Martinez, Gov. Haley or even Sen. Rubio come to mind).
But other than Clinton, I doubt any of the others have started serious vetting.