Over the years there have been many rivals. Most notably, the Colorado Avalanche were a strong rival, largely due to a cheap shot made by Claude Lemieux on Kris Draper (of the Wings) following a playoff game which shattered Draper's jaw. From that time forward, the teams played with intensity that approached hatred.
But in spite of any short term rivalries, there are five other teams that will always be strong rivals and those are the other original six teams. Of those five, the two nearest teams are strongest. There is a particular rivalry between Detroit and Toronto, particularly because Toronto is a Canadian city and hockey is a Canadian sport. But due in part to both teams continuing to be in the same conference, the strongest rivalry would have to be with the Chicago Blackhawks, also an original six team.
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Simple answer? The Montreal Canadiens. This dates from the formation of the National Hockey League in 1917 (Toronto & Montreal were original members). More recently, the Ottawa Senators have become a fierce rival, as Ottawa has a definite advantage in regular season play, but the Leafs have won every playoff series.
Throughout the history of the Flyers, their main rival has clearly been the New York Rangers. The "Broadstreet vs. Broadway" rivalry has been the main rivalry of the Philadelphia Flyers, since their creation. A close second would be with the New Jersey Devils, followed by the Flyers cross-state rivals the Pittsburg Penguins.
The Avalanche and Red Wings have what many say is the most heated rivalry in the league