As I will explain below, there is something like a 98% chance that Hillary Clinton will be the Democratic Party's nominee for President of the United States. You can safely ignore all developments in the remainder of the Democratic Party primary season and not miss anything.
Pay attention to the GOP race between Trump and Cruz instead, where there is at least a ghost of a chance of something interesting happening, even though Trump is overwhelmingly favored there as well.
The Results So Far
Clinton needs 17.4% of the remaining delegates (207 more) to win the Democratic Presidential nomination.
She leads Sanders in pledged delegates 1666 to 1359 (55% of those awarded so far) with 1,016 pledged delegates left to be awarded. A majority of the pledged delegates is 2,021, a threshold which she needs 355 more pledged delegates to surpass (about 35% of the remaining pledged delegates). Sanders needs to win 65% of the remaining pledged delegates to win a majority of the pledged delegates which he needs to make a moral claim that the super delegates should shift their support to him (in the absence of some new scandal impairing Hillary Clinton's prospects that provide an alternative basis to make this claim).
Clinton needs 720 delegates to have an outright majority of all of the delegates in the convention before even considering a single super delegate. This is only 84 delegates more than she would get in the worst case realistic scenario for Clinton that I outline below, in which she gets at least 636 more pledged delegates, something that could very conceivably happen or at least come very close to happening, if she wins 40%-45% in states where Sanders is favored (instead of the 20% that I've assumed) and 70%+ in states where she is a strong favorite (instead of the 50% that I've assumed) - which would win her about 67 more delegates than I have assumed in a realistic worst case scenario for her.
She leads Sanders in super delegates 510 to 41, with 174 super delegates still undecided at this point. She has endorsements from more than 92% of the already committed super delegates and from 70% of all of the outstanding super delegates.
All Democratic primaries and caucuses in the Presidential race allocate delegates proportionally.
The winner will be decided in the first round vote at the Democratic National Convention, because only two candidates are winning.
Clinton has gotten at least 20% of the vote in every state except Vermont, where she came close.
Remaining Races With Predictions
The remaining Democratic Presidential nomination races are as follows (with polling aggregated per Real Clear Politics when available):
Indiana - 83 delegates - Clinton 49.6%, Sanders 43% (Clinton expected delegates 41)
Guam - 7 delegates (Clinton expected delegates at least 2)
West Virginia - 29 delegates - conventional wisdom favors Clinton in West Virginia (Clinton expected delegates at least 15)
Kentucky - 55 delegates - conventional wisdom favors Clinton in Kentucky (Clinton expected delegates at least 28)
Oregon - 61 delegates - conventional wisdom favors Sanders in Oregon (Clinton expected delegates at least 12)
Virgin Islands - 7 delegates (Clinton expected delegates at least 2)
Puerto Rico - 60 delegates (Clinton expected delegates at least 12)
California - 475 delegates - Clinton 49.0%, Sanders 42.3% (Clinton expected delegates more than 238)
Montana - 21 delegates - conventional wisdom favors Sanders in Montana (Clinton expected delegates at least 4)
New Jersey - 126 delegates - Clinton 51%, Sanders 42% (Clinton expected delegates more than 63)
New Mexico - 34 delegates (Clinton expected delegates at least 7)
North Dakota - 18 delegates - conventional wisdom favors Sanders in North Dakota (Clinton expected delegates at least 4)
South Dakota - 20 delegates - conventional wisdom favors Sanders in South Dakota (Clinton expected delegates at least 4)
District of Columbia - 20 delegates (Clinton expected delegates at least 4).
Minimum expected additional Clinton delegates assuming 20% where polling is not available and she is not clearly favored by conventional wisdom, 50% where polling is available and she is favored by conventional wisdom, and per her polling assuming all undecided voters go to Sanders where polling is available: 636
This is basically a worst realistic case scenario. In it, she is almost sure to win both a majority of the pledged delegates and a majority of all of the delegates by a wide margin.
This prediction is highly robust because the assumptions made are very conservative and because the proportional delegate allocation rules make huge shifts in the popular vote necessary to cause significant shifts in the delegate count, especially when so many of the delegates have already been pledged, buffering any shift in the late polling that could influence the number of delegates awarded.
In this scenario, she can be abandoned by 84% of her super delegates and still win the race. And, there is no reason to expect that even half of her super delegates will abandon her in favor of Sanders if Sanders does not win even a majority of the pledged delegates, which he will not.
In reality, Hillary Clinton will almost surely win more than 636 pledged delegates in the remaining races, and could afford to lose even more of her super delegates and still win. Since Clinton's super delegate advantage will be a non-issue, the legitimacy of the nomination process will be untarnished, and Sanders will almost certainly acknowledge defeat and urge his supporters to back Clinton in the general election.
The odds of Sanders winning the Democratic Presidential nomination is much less than 1% in the absence of a major new scandal implicating Hillary Clinton, or a major deterioration in Hillary Clinton's health (e.g. an assassination prior to the Democratic National Convention like the one that killed Robert Kennedy). The odds of either of those things happening is at most 1-2% or so in my estimation.
The odds the Hillary Clinton will be the Democratic Presidential nominee is on the order of 98%.
The Sanders campaign has basically conceded this mathematical reality, has laid off hundreds of staffers as a result, and is now focusing on putting together a large slate of delegates to influence the party's platform and spreading his message, rather than on winning the Presidential nomination itself.