In today's salary-cap driven NHL, parity is the name of the game. More than ever before, the playing field has leveled out. As we've seen in the playoffs, almost anything is now possible.
Unless there's a fundamental change in the way the NHL does business, it's not likely that we'll ever again see teams dominate for years on end like the Edmonton Oilers did in the mid-to-late '80s, the New York Islanders did in the early '80s or the Montreal Canadiens did in the late '70s—and in many prior eras.
What makes a modern dynasty? Here's Rory Boylen's suggestion, from a 2011 article in The Hockey News:
We shouldn’t get so lenient with the definition of a dynasty that we automatically have one each decade, because that would water down the feat. But if a team is consistently dominant and has at least one championship, it’s fair to put it in that pantheon. It may not be the greatest ever, but it’s still up there.
The NHL currently boasts a few consistently good teams, some of which are starting to satisfy Boylen's criteria. To be in consideration right now as a future dynasty, I'd say a team needs to have reached at least the second round of the 2013-14 playoffs, have collected some championship hardware and look like its future is bright.
Here's how last spring's eight conference semifinalists rank in terms of the likelihood they'll reach dynasty status over the next few seasons.