Been a Ranger fan since 1970 and as
@Tawnos quite adeptly layed it out, Rangers have pretty much have had as much sustained "success" if success is quantified by playoff appearances and not titles as any franchise in any sport!.
That said , the most memorable for me years for Ranger success was:
1971-72 Rangers were an absolute powerhouse. Now I'm just 9 years old and tho I'm playing ice hockey at Abe Stark rink as a kid in Brooklyn and learning the nuances of the game, I don't really truly get the significance of that team as I did as I got a bit older. Don't forget there is no cable back than and the only games you could watch were road games in black and white TV or if you were lucky, the national game would be a Ranger home game.
That team should have won a Cup tho. No question in my mind and many other Ranger fans who were around at the time but had Jean Ratelle who was on his way too a 60 goal-85 assist year and that's not a missprint, not broken his ankle, Rangers would have beaten the Big Bad Bruins even with Bobby Orr at his magical peak. But it was not meant too be as hockey gods conspired against us and we came up short in the finals. But that was a fun fun year too be a Ranger fan.
1978-79 Rangers with Cup run from out of nowhere. Honestly tho there are alot of us Ranger fans who believe we won our Cup by beating the heavily favored Islanders in the ECF. The Habs were still on their dynasty run and even with a healthy Davidson, we still likely dont win the Cup that year. But man oh man....that was about as fun too be a Ranger fan as any season was.
1983-84 Rangers which was the best of all those early 80's Herbie Brooks coached teams where we camethisclose to beating another heavily favored Islander team when we lost on the Morrow goal....that was a really fun team too watch. It was basically the smurfs and Barry Beck but that teams was really good and had they beaten the Islanders I really believe that team could have and would have at worst made the Cup finals if not have won the Cup that year...lol
1985-86 Rangers which IMO is the most underrated team we ever had as no one ever talks about it. But that Ted Sator led team finished year under .500 sneaks into playoffs and than out of friggen nowhere again, beats heavily and I do mean heavily favored Flyers and Caps teams to get too ECF where eventually the Habs beat us. That playoff run was one of the most fun times I ever had as a Ranger fan.
1991-92 Ranger team should have won the Cup that year...I firmly believe it. I still think that team was better than the '94 Cup winning team. Hockey gods conspired against us as goals from center ice somehow go in. Graves gets suspended which hurt hugely. That all said, that was a fun fun year too be a Ranger fan.
1993-94 Cup winning Ranger team was ridiculously fun for obvious reasons...enough said...lol
1996-97 Ranger team was also alot of fun too watch. Gretzky, Messier and Leetch with one last hurrah too get another Cup but getting beat by the Flyers in the ECF was tough too take. But not just beating but smoking some really good Panther and Devil teams before that was so friggen enjoyable tho that it was a memorable year.
2011-12 thru 2014-15 were just about as fun a 4 year consecutive run as any Ranger fan could have possibly wanted or hoped for other than the fact we didn't win a Cup. Every spring as weather got nicer and days longer which I personally love and is my favorite time of year, we all had 2 months of amazing, stressfull and memorable playoff games too watch. We can only hope that whatever new form of Ranger teams emerge as this rebuild/retool comes to it's conclusion soon, can give us the excitement that those 4 years gave us.
I'm pretty confident it will but until it actually happens, these are the seasons that IMO are the most memorable ones in my Ranger fandom.