Keep in mind that I've only ever played brutal (except when doing the mastery achievements, I went to hard for many of them) so when talking about the design I'm only talking about the "brutal" version, and I'm only talking about the main and bonus objectives, not the achievements. I get a little long winded and go into a lot of detail on why I love these missions. By no means am I anything close to an expert on game design, so a lot of these details and reasons are really just why I personally like them.
1 - Supernova
This is the banshee mission in WoL. It may well be my favourite mission in all of SC2. Everything about it is just perfect in my opinion. The units introduced are very helpful for the mission, but not mandatory (unlike in some missions), which is what I want from a game that introduces a new tool. I like to know that all the tools I collected throughout the game have good use in a place that's not limited to where I found it, but also very helpful but not necessary for where I did find it. The banshees are helpful, of course, because many of the bases that you're trying to seize is guarded by minimal ground vs air (cannons, stalkers, and sentries) while the bulk of the power is stored in heavy anti-ground (immortals, colossi, etc.) Though late in the mission there's quite a bit of HT and archons, so if you're not careful, you can lose all your banshees to storms.
For me, the urgency created by the fire wall is such a clever game design mechanic. There are missions in SC2 that say things like "complete X before this timer runs out", which is a lot more of an artificial sense of urgency that the characters in the plot care about, not you, the player. The idea is "if the clock runs out then the field generators of this ship will stop producing power for our main defense cannons against Amon" or some such nonsense that you don't really care about because you're in the middle of playing a game. But in this case, "if you don't move your buildings to the next base fast enough, your buildings will burn to death!" As a player you care about this tremendously, and that creates such a natural tension of watching the fire wall get closer. And all the bases are heavily guarded. The smaller bases that aren't far enough to the right aren't guarded as well, so you have to make an on-the-fly decision about whether or not it's worth the effort to take the small bases. This is such a great way of forcing the player to make a value judgement. And while the bonus objectives are slightly out of the way, they're not so heavily guarded as to completely distract you from your primary purpose. To me the challenge of having to push forward on shaky grounds, the natural tension created by the fire wall, and the utility of banshees just makes this such a perfectly designed mission I absolutely love it.
2 - Ghosts in the Fog
This is the second LotV prologue mission, and probably my favourite original mission design in LotV. This is the one where you have to break into a facility, but you don't have any vespene geysers; instead natural refineries periodically spew gas pickups throughout the map, but they're all guarded. I love this mission because of the clever back and forth balance between the need to push further in the map so you can have resources to build better units, and the need to have more resources to build better units so you can push further in the map. It's not just the gas, the only other two mineral bases you can get are really heavily guarded, and you need to have already collected a ton of gas to get those.
This is one of my favourite scarce resource missions in SC2 because it really clearly defines, more than any other mission, the challenge of how you're going to spend your gas vs how you will dump your minerals. And the process of actually "harvesting" that gas is so heavily tied into the actual gameplay of pushing out and attacking bases that you actually feel like you're playing the game by mining. This is opposed to the standard SC2 model where mining is a completely separate task that you tell your workers to do, and then forget about while you're busy with the more "important stuff". I think that's a really clever way to tie the resource game into the actual game. I also love how challenging it is; it feels like one of the hardest missions in LotV. I only wish that they put a clever mission like this in the actual game itself, where you have already collected such kickass tools that you want to figure out how to use in this map, but of course made it much harder to compensate. And while this is, in principle, a mission with no countdown clock running out, there's a very natural countdown that creates urgency, which is the fact that you're running out of resources, and if you don't make something happen soon, you're screwed. Again, it's always better to create a natural sense of urgency than an artificial one.
3 - Piercing the Shroud
This is the secret WoL micro mission, where you play Raynor infiltrating the hybrid lab with a bunch of marines and medics. This is my favourite micro mission in all SC2. It's really hard and fast paced, because most of your units are so fragile but it's extremely important to preserve all of them. So you're trying to tank everything with Raynor as much as possible, and it's always a difficult question to choose whether or not to stim in this scenario, because that 10 HP can make a huge difference. However, what I think is mostly so clever about this mission is the weapon pickups that Raynor gets, and how well they tie into the mission. They're very straightforward. You get ammo for making one huge single target shot, or a big AoE shot. And the mission is so well designed in a way that you feel like you need them very often. Because very often, the things you're about to fight is way too strong for just a few marines and medics. There's also three "switch points" where you get to make a decision (which things to unleash on the enemies, and what reinforcements to get) but that's a little gimmicky, so I while it's fun, I don't consider it a highlight. It's certainly not a consistent mechanic cleverly designed to tie well into the mission like the consumable pickups are.
The best part of this mission, though, is the end. I always love, in video games, that moment where you're being chased by something that's so substantially powerful that engaging it is absolutely out of the question. But it's still hunting you aggressively, and all the obstacles in front of you make it difficult to escape the thing. Like with the fire wall in Supernova, this is a very natural way to create urgency in a game, without doing something gimmicky like making a countdown timer that the characters care about, but not the player. You and Raynor care about getting away from that giant monster for the exact same reason - YOU DON'T WANT TO F***ING DIE! And then, this gives the best consumable pickup in the game, the chrono rift! While there are so many obstacles preventing you from escaping the hybrid, there's also great strategic vantage points to use this thing and help you escape. And it's almost mandatory, especially if you want to get the last bonus objective.
4 - Salvation
I absolutely love defense missions, and I had to include one. It's a tough call between this and "All In", but I went with this (mostly because I don't want my list to be too WoL-centric). This is the last LotV mission before the epilogue, and in my opinion, the hardest LotV mission including the prologue and epilogue. There may not be anything very clever about it on the surface, it's a straightforward case of "defend a bunch of attack waves". But it is just so mechanically sound as a defense mission. Unlike the earlier defense mission in LotV, which revolved around defensive buildings and such, this one doesn't revolve around that. While it is extremely important in this mission (more so for the sides than the front) what it really revolves around is a clever use of all the tools you collected throughout the campaign. If you just do a straightforward strong a-move army approch + cannons to this mission, you will probably get crushed. You have to use things like the arbiter stasis field to hold waves at bay, well placed anti-ground AoE for the massive ground swarms (things like reavers), etc. And you have to make very quick decisions about where you'll send what units at which time, because simply creating a unit that's "assigned" to a side for the rest of the mission absolutely won't cut it. And yet, even though you often probably feel overwhelmed and that you can barely make it, there's still a ton of strategies for beating it. You can take care of the armada ships with dark archons, wrath walkers, tempests, or other such stuff. You can either make heavy anti-ground for the massive ground waves, or mostly divert those resources for other things while you rely on spear calldowns or stasis fields for those waves. I try to separate my thoughts on a mission design from the between-mission mechanics of the campaign, but I think that the tools you get throughout the campaign just manifest themselves so perfectly for this mission. Again, there's nothing innovative that pops out in the mission itself, but it's just such a solid, perfectly designed defense mission that it's way up there in my book.
5 - The Dig
This is the siege tank mission in WoL. It's another defense mission, and probably my favourite one that isn't a "final" mission (I still like All In and Salvation a little more). Of course, the laser is what makes this mission. Because it takes so long to change its target, you really have to make the value judgement of whether or not killing a certain unit is worth prolonging the mission (because the longer it goes, the stronger the attack waves become). I really love the mechanic, it just feels so good to use, and it's such a great way to make you multitask in a defense mission and feel like you're doing something for real. They could have made the laser drill a pure storyline tool, but instead, they let it interact with the player and their goals, and I like that. I also really love Essence of Eternity, the second epilogue mission which basically used the same design.
I guess you'll notice that there's no HotS missions in the list. Oddly enough, zerg is my best race, but HotS was the weakest campaign in my mind, in both mission design and "leveling up mechanics". Now HotS did have some great missions (though most of the innovation seemed to come in the micro missions in that campaign; none as good as Piercing the Shroud IMO). In fact, there were some that I thought had enough promise to make my list, except that the execution wasn't that great.
For example, what I thought was the most promising mission idea was the infestor mission, with the virophages where you have to bust into Narud's hybrid facility. The whole mission was about holding on to your virophages and keeping them from being attacked so that you can have more reinforcements when the fog is released. However, because of how OP infestors and Kerrigan are, you don't even really need any of the virophages to bust into the entrance, so the easiness of that mission kind of nullified the point of its design. One way it could have been made better was to improve the strength of your infested terran reinforcements from virophages (and possibly make it an accelerating return, rather than linear with the number of virophages you have) but also make the terran defense substantially stronger, and constantly replacing units. They could have done this in such a way that there is absolutely no way you have a chance in hell of busting that door on your own, and you absolutely need the phages, probably at least 3/4 of them at all time.