Not sure where to start. First of all, I've only used soft(ish) cups (rated 2-4 here) and if I avoid trapping my (very low) cervix in the folded part, I've learned to make them open pretty much at once.
What I've personally found most useful is the cold water trick, as originally described here. It honestly isn't anyhow uncomfortable, though I sometimes rinse my hands with warm water after that:) I can't stress how much I recommend trying it out. Be sure to pour the cold water over the outside of the cup and to do it for long enough - the original post recommends 10 seconds but basically your cup just should feel a tiny bit stiffer. (Just my personal opinion: if in doubt, get a softer cup. You can make a soft cup stiffer for insertion but you can't make a stiff cup softer for more comfortable wearing).
If you happen to use your cup inside out and it's less springy, what works for me is to turn it right side out (i do it anyway when washing), run it under cold water, turn it inside out and after running a little under cold water insert asap. It's as if it remains springy when you've just turned it inside out:)) (Warning: turning a Ladycup inside out can cause the holes to damage. I never needed this with it though, only with my medium Meluna Soft.)
As a low cervix gal, I also have to let any cup open as soon as the rim is in, and I've only recently realized that it might make a huge difference too. As with everything concerning cups - experiment! And by the way, I think that's an overlooked reason to get a softer cup - if you have difficulties making your cup pop open because your cervix is in the way, a stiffer one will indeed open regardless of that - and from what I've heard, it will be ouchy. Another reason not to rush and buy another cup.
So, make sure your cervix isn't in the way and if it is, move the cup a bit lower. No idea how this works but pinching the middle also seems to help me get my cervix into the cup, especially with my medium soft (glittery) meluna.
Many have already heard those other suggestions, but I'll reiterate on them.
The fold makes a huge difference. Experiment, face it this or that way (especially towards your rectum where there's no bone in the way). Also keep in mind that the air has to go in either above the cup or through the holes. One trick I've heard is making sure one of the holes is in the middle of the crease, especially for the punchdown fold. (mostly out of curiosity - any other tricks related to the holes? maybe cup-specific, like with the fleurcup's unusual placement of them?) Pay special attention to this if you use a Meluna Soft, as Melunas have only two holes, not four. Also, make sure the holes aren't clogged with blood (especially when inserting without rinsing) or lube.
By the way, if your cup is a Diva, you may want to enlarge the holes - frankly speaking I don't know how, ask around!
I also recommend trying out the "9-fold". I've heard of it here but never saw it, but this is what I myself call so:
Basically, an incomplete C-fold. That's what works best with my large Ladycup inside out, the softest thing ever. It's a pretty wide fold, though. The folded part may or may not be a little lower than the rim.
Another fold to try out is the 7 fold/triangle. The origami fold is also mentioned often, especially for the Ladycup. You can unfold these manually. (a great video on folding techniques)
Hopefully you've also already heard the advice to push away the wall of your vagina in order to let some air in. That's right, you need to get air in/above your cup. When you do that, push away the back soft wall and make sure to hold it like that for long enough. 15-20 seconds maybe. You may hear a sound as the air goes in.
The general advise about relaxing is also important here. When your muscles are tensed up, there's less room for your cup. Keep your jaw slack (no chewing gum!), remember to breathe, relax your back and shoulders. Try "blowing raspberries" :D You've probably heard of kegel exercises already. When you've identified these muscles, learn to RELAX them as well.
As far as I understand, whatever is advised during childbirth should work too.
Anyone who's read this far deserves a little reward:) Cup icons!
The three in the middle are based on a photo by whimaway, originally posted here. have a look, really the only photo I've seen so far that really does the glittery Melunas justice <3 (by the way, these were in my entry for free20in20. nice way to promote cups, methinks).
Back to serious business, suggestions to improve this post are very welcome. To those of you that use softer cups - which of these tricks do you find particularly helpful? any other tricks/tips? If you don't need to do anything special, also say so, this might encourage someone else <3
If you are new to cups and a struggling with a soft one, remember that other cups may have other difficulties about them, so don't give up too soon! The learning curve is also about figuring out what's the best way for you to use a cup.
Also, I tagged this post with some cups this applies to, but please note that I only have personal experience with the small Lunette, large Ladycup, large Green Donna (dry run) and some soft Melunas. I'll be happy to speak about my experience with them but consider asking in the community itself too.