TL;DR: Probably President Clinton. Well, it was already the most probably outcome before any primaries happened, but it's getting kinda close to near-certain now. Here's how I see the numbers...
Bernie Sanders started his campaign with a solid, realistic chance at getting nominated, but a small one. Beating Clinton was always going to be very difficult for him, though definitely possible. He made a valiant effort, and I hope he continues at it for a couple more months, but unfortunately it looks like it's not happening. Clinton has won unless something big and unexpected happens.
Before I say why I believe that, let's get one thing out of the way: Forget the superdelegates. They don't matter. Whoever wins more pledged (elected) delegates will get the nomination, period.
Why? Imagine if one candidate won a majority of pledged delegates, but superdelegates pushed a different candidate over the top to get the nomination. We'd have a contested convention, because people wouldn't know who the nominee would be until the convention decided, which means that the process of supporters of the losing candidates getting behind the winning nominee, which normally happens before the convention, wouldn't get started until after the convention - too soon before the election. Worse, if the candidate who won more delegates in primaries didn't get the nomination, a lot of that candidate's supporters would feel, quite rightly, that the nomination was illegitimately stolen from them. It's a recipe for losing the election.
Superdelegates know that, and they're party leaders - the very people who most care about the good of the party. So they won't let that situation happen. When we know for sure who's going to have a majority of pledged delegates, superdelegates will switch their support if necessary to make sure that candidate gets the nomination. That's what happened in 2008: Clinton started with a very large superdelegate lead over Obama, but near the end of the primaries most of the undeclared superdelegates declared their support for Obama, and many of the Clinton superdelegates switched to him.
So, superdelegates are irrelevant. Just look at the pledged delegate counts.
Predicting election results from polls and intuition and demographics and experience is one thing. We're at a different stage: Now that Super Tuesday has happened, about 1/4 of pledged delegates have already been voted on. Looking at actual voting results in a primary contest is usually a very reliable indicator of future voting in the same primary contest. We can see the patterns, and have high confidence they won't change much as long as the same candidates are running.
What do we see? First and foremost, we see Clinton leading by around 200 pledged delegates. With a significant chunk of delegates already voted on, that means only 3/4 of delegates remain to be elected, and Sanders would need to make up that ~200 delegate deficit to win.
Where is Sanders going to gain 200+ delegates over Clinton? I can't see a way for him to do it.
Lots of people point to the fact that he had big wins in Colorado and Minnesota on Super Tuesday, in addition to his home state Vermont. This indicates, they suggest, that he can win some of the bigger states coming up, outside the south. But I see something different: Colorado and Minnesota were the only caucus states on Super Tuesday. What I think we saw is that Sanders' campaign is getting very good at caucuses. Which is great, but not enough.
This is actually a big part of how Obama beat Clinton: By beating her solidly in nearly every caucus state, he racked up slow but steady delegate leads to offset some of her gains in the bigger states. Coupled with very strong wins in southern states with high black populations, that allowed him to maintain a modest delegate lead she was never able to quite match.
Problem for Sanders is, Clinton is winning overwhelmingly in the southern states! We saw that on Super Tuesday and should expect it to continue. Caucus states are not numerous enough, and most of them are small; caucuses + rural northern New England won't be enough to even break even with Clinton, let alone overtake her. He needs something else.
Massachusetts is what showed us that he's not likely to get much else.
Yes, sure, it's a remarkable achievement for him to have pretty much broken even with Clinton in Massachusetts. It is indeed impressive, and it shows that his campaign has a lot more support and strength than most people foresaw. Clinton's strength is in states with a strong Democratic establishment, people with a lot of party history who have connections to the Clintons. Places where the cities big and small have strong Democratic parties with Clinton-supporting mayors and aldermen and councilors who have networks of people they can turn out to vote. Clinton should've dominated here, and instead, the result was so close that Sanders and Clinton will each get the same number of delegates.
But impressive as it is, it's not enough. If Sanders cannot win Massachusetts, he's going to do worse in the states with more established party organizations, or even similar ones that are further from Vermont. He's going to lose NY and CT and RI and CA and NJ and MD... he's going to fall further behind on delegates. Where can he make up for it?
Sanders' campaign is suggesting the midwest, but if he could win solid wins in midwestern primaries, he'd have been able to win solidly in Massachusetts too. Michigan is coming up soon, and every poll there so far - including a few within the past week - show Clinton up by double digits. Maaaybe Sanders can close the gap, but what he needs is a big win. Instead, it seems the only question is whether Clinton's delegate lead will grow just a little bit, or a lot. Other midwestern primaries will follow a similar pattern. In the meantime, Clinton will continue gaining large delegate leads in primaries outside the midwest, with the south cancelling out Sanders' gains from caucuses.
Edit: Since I first wrote this, we got results from several more states that happen to perfectly illustrate what I'm saying.
Sanders solidly won caucuses in Kansas, Nebraska, and Maine. Clinton overwhelmingly won a primary in Louisiana. Net result: About the same number of delegates for each, with Clinton's lead remaining at around 200 delegates.
Kansas: Sanders 23, Clinton 10
Nebraska: Sanders 15, Clinton 10
Maine: Sanders 16, Clinton 9
Louisiana: Sanders 14, Clinton 37
Total: Sanders 68, Clinton 66
This is actually quite representative of the south vs. caucuses in general. Sanders is not going to make up his delegate deficit this way. What remain, mostly, are the high-population coastal states and midwestern primaries. Sanders could win significantly in Oregon and Washington, but in the rest it looks like he will either lose or break even, meaning that overall Clinton's lead will probably grow, and certainly not shrink by 200.
Sanders should keep on running. Why?
- Clinton might stumble, or some big unexpected thing might happen. It's out of his control - Sanders can't make Clinton lose. But it's always good to still be there, still running, if some shock upsets the race, so he's in a position to pick up the pieces. Not likely, but if it does happen, it's good to have a strong candidate ready as a backup to take the nomination.
- Contested Democratic primaries and the attention paid to them will help Democrats win in November. More Democratic debates, contrasting with the ridiculous Republican spectacle, help - but they won't help much if Sanders stops running and people stop watching them. The appearance of a real contest in each state about to vote is also going to give a lot more press coverage to Democratic issues, to offset press coverage of Republican issues and prevent those from dominating people's attention. And when Clinton and Sanders organize for primary votes, they're building organization for the general election too. More Democratic organizers will gain experience, connections with voters, and familiarity with this year's neighborhoods and polling places and reporters and so on.
- Sanders' challenge makes Clinton a stronger candidate. As she adapts to him, she becomes better able compete against others. That's especially true if Trump gets the Republican nomination, because he'll hit Clinton at her weakest point: the idea that politicians are bought and paid for and work for the corporations and lobbyists who buy them. Sanders has been hitting a similar theme, and she's learning how to respond to that and in the process developing the messages she'll need to fend off Trump.
In the 2008 primaries, when Obama's victory was clear from the numbers but Clinton kept campaigning for several more months, we had a wrinkle that worked against my usual recommendation that contested primaries are good. Obama's lead was modest and Clinton had the support of a lot more superdelegates, and various Clinton campaign sources and others were suggesting that she should try to contest the convention even if Obama won the primaries. Although I doubt she could've done so successfully, the mere attempt would've been very destructive to Obama's chances in November... and Clinton kept refusing to disavow such plans! In the face of that, it was important to ensure that Obama's delegate lead would be large enough to dissuade Clinton from even trying to contest the convention, and that's why at the time I urged everyone voting in the primaries to vote for Obama even if they preferred Clinton.
But we have no such wrinkle this year. There's no prospect whatsoever of Sanders attempting to contest the convention if Clinton wins a majority of pledged delegates. So the better Sanders does in remaining primaries, the better off we'll be. Especially in the extremely unlikely case that he pulls off a victory - a legitimate one.
I used to say Trump had no chance at the nomination, and I've changed my mind, but I want to tell you why: Trump is a sure loser for the Republicans, and most Republican leaders know it. Not only that, but he's antithetical to much of what they believe in. So unlike anyone else in a multicandidate race, it's not enough for Trump to win more delegates than any one of his rivals. To be sure of the nomination, he needs to win a majority, that is, more than all of his rivals combined. If he goes into the convention with less than a majority, Republicans will be very tempted to prevent him from being nominated.
Yes, as I said above, a contested convention is a recipe for losing the general election. But nominating Trump is also a recipe for losing, perhaps even more surely. So in Trump's case, a contested convention may not be the worst option.
However, what I didn't consider was the effect of winner-take-all states. I knew that in the Republican primary, some states are winner-take-all (as opposed to the Democratic primary, where they're all kinda-proportional). It turns out to be a lot of states, including some of the biggest. Including Ohio and Florida, for example. And there's a chance Trump could win them both, and get all of their delegates. So... it looks like Trump does have a chance at an actual majority.
Even if he doesn't get a true majority, if Trump wins enough of the big winner-take-all states, he could go into the convention with a very large lead over every other candidate. That would greatly increase the risk to the Republican party of not nominating him, since it would seem so very unfair to his supporters. So they might let him have the nomination.
Super Tuesday made that more likely, because Marco Rubio did just well enough to ensure he wouldn't drop out. He'll stay in long enough to try to win Florida. Whether he wins or loses there, it means enough delegates will be split between him and Cruz in the states in between, that Trump could keep a very large lead if he wins some winner-take-all states. When it comes to the convention, Republicans will be faced with a really bad lose-lose choice: Trump or contested convention.
Whichever of those they go with, chances are, Clinton will beat them in November.
Edit: Since I first started writing this, we've also had a few more Republican primaries. We see that Trump's lead is declining a little, but that he could still win Florida or Ohio or both. The chances of a contested Republican convention are increasing, especially if Trump doesn't win both of those states.