What a glorious topic! This is one to get a cup of tea and a plate of biccies for...

As far as classic skills go, I never really played
Lemmings 2 or any of the games that followed it - my only real experience of Lemmings came from the original game and
Oh No!. Thusly, skills like the Walker, Jumper, Disarmer, Swimmer and Cloner all feel completely novel to me and, whilst I love them all, I wouldn't consider them to be classic-style skills.
The Stacker, perhaps, could be considered classic because of its similarity to a Builder and its interchangeability with a Blocker. However, ironically I wouldn't include it in a "classic" skill panel for much the same reasons!
If you had to choose just 10 skills to build levels with, which would it be? This is a distinct question from "which 10 are the most classic"; it's about which 10 work together the best and feel the most cohesive as a skillset for an entire game.
Brilliant question.
My Main 10 - I'd be happy using these as the exclusive skill panel for a level pack:
Walker - This has become one of my favourite skills. It has a multitude of uses, and is also great for speed-solving purposes. It's backroute prone, sure, but with the right application it can be the most powerful skill on the panel without opening up backroutes at all!
Climber - For much the same reasons Proxima said (gaining height, separating a worker, etc), and particularly its up-and-coming interaction with Jumpers.
Swimmer - I
completely agree with Strato here: the Swimmer is a great skill for way more than just getting across water. You can use it to separate a worker, make a drop safe, gain height (in quite a unique way); also check out my use of the Swimmer in the
new-skills-challenge version of
ROCKY VI - it's the key skill that made this solution possible! Also, it makes solving
Hunt The Nessy a breeze!
Glider - This was a tough one, because I prefer the way the Floater
looks with its red and white umbrella (I guess I could just customise the sprite

), but the Glider is just
so much more useful.
Blocker - Beats the Stacker and Stoner hands down as an instant way to hold back or separate a crowd into sections; also useful for turning workers, and they can be rescued.
Builder - For all reasons already mentioned, it's simply an indispensable skill with power and puzzle potential.
Basher - This is probably the most tentative addition to this list, but earns its spot simply because it's too useful not to. Bash-cancelling can be a great way to gain height through a block of terrain, plus it's an instant way to get through thin terrain without sacrificing a lemming with a Bomber. That, plus its worker-separating and safety-pit potential gets it past the mark. Just!
Miner - Another definite; instant way to remove a couple of pixels of ground, you can use it to rescue Blockers, get past traps, create safety pits, make your way downwards through terrain whilst also creating a safety drop, and it responds to Blockers. You can also use it to turn lemmings around as well. Proxima's declaration of it being the "King of the destructive skills" is, I feel, no exaggeration.
Cloner - I just love this skill. Its puzzle potential for making a lemming perform the same task in two directions whilst only using one of the skill initially is very powerful. It can also be used to create a worker lem facing the opposite direction, and it opens up multiple-solution possibilities.
Jumper - It's not been officially released yet, but what a skill this is turning out to be! I can already think of a handful of levels in
Lemminas that I'd like to use this skill for, and there may well be another release version of the pack once this skill has been fully implemented into NeoLemmix. I think that fact alone puts it quite firmly on my list.
Guest List - I'd feature these as guest skills in a pack with otherwise just the above panel; these are all skills I use extensively:
(in order of preference)
Disarmer - I will always feel an affinity for the Disarmer, having created the currently-used sprite for it, and it makes a valuable pick-up skill as well as having more puzzle
and even action potential than people give it credit for (especially at high release rates or other tight-crowd situations). This would be a regular guest skill, for sure.
Platformer - A great alternative to the Builder for gaining ground horizontally whilst restricting vertical movement, but also great for quickly closing gaps, making safety platforms, and building underneath low ceilings. It was very difficult not to put this on my main list, but it's just too close to the builder, ultimately. A worthy guest skill.
Shimmier - A great skill, but the Jumper, Walker and Swimmer are all better alternatives for its main uses. I do enjoy creating Shimmier puzzles and levels though, so this is one I'd miss if it weren't there at all.
Floater - Again, it was hard not to put this on my list, but I generally do perceive the Glider as being more useful for getting lems down from a height, and the Floater is the more reactive of the two skills. I'd probably either use it as a guest skill or assign it to the occasional entrance hatch.
Bomber - As much as I love all-save solutions, I have to admit that the Bomber remains the secret best of the destructive skills for its versatility, ease and quickness of application, and universal assignability. With NeoLemmix's removal of the timer to boot, I'd definitely use it as an occasional guest skill, albeit sparingly. And, I'd just have to use talismans for all-save versions of my Bomber levels!
Stacker - This one only just makes the Guest List, and then only for its special-trick potential. It can be useful for creating vertical terrain for Climbers, but most of its other main uses can be covered by Blockers, Builders and Platformers to slightly better effect. The less obvious Stacker tricks are what earn it this last spot in my Guest List; it does occasionally feature in my levels as one of the tougher puzzle elements (see
With A Twist Of Lemmina Please and
Milkshake Party, currently rated as the most difficult levels in the
Lemminas pack and both of which use the Stacker as a main part of the puzzle!). Also, I recently discovered a trick involving Stacker>Climber>Jumper thanks to
one of DireKrow's Jumper levels which could prove to be very nifty! So... I wouldn't ever want to see it disappear, but it would be a guest skill I'd only use very sparingly.
Skills I Wouldn't Miss - I acknowledge that these skills have their use, and as a result of this post I might start looking at ways to use them more creatively, but for now they're off the list:
Fencer - Easier than bash-cancelling to gain height through terrain, granted. Buuut I dunno, there's something about the Fencer that, whilst I use it regularly (and probably favour it more than some of the skills mentioned above, ironically), I can't quite put my finger on why I wouldn't consciously count it as a go-to skill. I likely won't stop using it, and it's by no means my least favourite skill, but I think I could do without it.
Stoner - This is a skill I've never really liked, and I've never used it in a level. I can think of better ways to break a fall that don't require sacrificing a lemming, and neither Blockers nor Stackers are sacrificial either, which is the Stoner's other
main use. That being said, it
can be used to create terrain for a Swimmer to stand on (but again - it doesn't feel like this is worth sacrificing a lemming).
However! The appearance of the Jumper skill is the first thing to spark my interest in the Stoner: all of a sudden, you can create terrain
above the lemmings very quickly, and also give other Jumpers something to turn around on (thanks to the much debated hit wall-turn-fall mechanic of the Jumper!). So, whilst for now I'm pretty sure I wouldn't want it in my Main 10 or even my Guest List, I am beginning to see the skill in a new light.
And finally, the
Digger - What can I say... I just feel I could do without it it if I absolutely had to. Digger pits are a pain to get right, and there are more useful and entertaining crowd-control methods. Also, the vertical shaft they create is often lethal if left unchecked. Whilst these things are, admittedly, less of a pain in NeoLemmix than they were in the original game, they seem to have left me with a general disdain for the Digger skill in general. I'll use it if I need to, but its not one of my favourites.
So that's it! Looking forward to reading others' choices!
ALSO: As a possible extension to this thread, how about making a challenge to solve the original game (or some other level pack of choice) with only your chosen 10! I think it would be interesting to see how that turns out!