So far, I have three level capped toons and 3 more in the range of 69-73. They have all been very different and their soloability has been much different.
The first one I leveled was my prot warrior. I was decked out in all epics from the last expansion and had very little trouble with anything. I fact I soloed pretty much every elite mob on my way to 80 as well (Dungeon mobs withstanding). If I had been poorly geared, this would not have been nearly as easy. One other small problem is that before level 75, a warrior has to either rely on bandages and pots to heal in combat. Once you get Enraged Regeneration, it becomes that much easier. Obviously being well geared and prot has many advantages in this situation. You have tons of health and armor for starters. You also have a high amount of avoidance and dont tend to take much damage when you are hit. If you have the gear, I would advise prot leveling over arms or fury because of the amazing survivability.
My second toon to level cap was significantly different. It is a resto druid and was leveled a bit before dual specs came out. With a resto druid, you CAN solo things. It is just a long and painful experience. You damage output is very low and you are quite squishy as well. The one saving grace to leveling with this spec is that you can throw a bunch of hots on yourself. Hots allow you to do more pew pewing than would be possible if you were having to throw heals with big cast times. Most quests went pretty slow. If you are leveling a resto druid, I would advise to level them through instances (which I did a lot of). You can ALWAYS get into a group as a healer while you are leveling. My Druid actually had about half BC epics to start as I got into a Karazahn group at 68 prior to Wrath coming out (and was first on the healing meters against a t5 priest....go go nerf wild growth).
My third toon to hit the cap was my pally. All of BC, he was a flash of light spammer. Since I leveled my Druid, I decided to do him as Ret. My Ret set was as hodgepodge as it got. Several BC quest and dungeon blues, a few green items , and a few epic items that I managed to snag from Kara during my many flash of light fests in there. The biggest thing I had was Gorehowl. Having a nice weapon obviously helps. I picked up several new pieces of gear with the first few quests I did in Borean Tundra. Soloing quests was not really hard at all. If they were non-elite mobs I did not have any problems at all. Being able to heal yourself is a huge benefit and the Art of War procs were very useful. It was a big too much to solo some of the elites, but for the most part I could blaze through whatever was put in my path.
I am now in the process of leveling my Warlock (will be next 80), Deathknight, and Hunter. So far they have been very different. My Warlock can pretty much solo everything with his affliction spec. My corruptions heals me and I have haunt as well. I can basically run into a group of 6-8 mobs and kill them all without losing so much as a sliver of health. Obviously it helps that mych of my gear is Season 4 Arena gear. I have also soloed every mob I have encountered including the Prince at the end of the Thassarian quest line in Borean.
My Deathknight has pretty much been a faceroll fest as well. I am wearing pretty much all Northrend Quest items and smashing mobs to bits. I soloed everything except the prince which was two manned. I even soloed Harold Lane despite his annoying snare tricks.
The last of my toons to hit Northrend has had a lot more difficulty. My Hunter entered wearing Outland quest blues and greens and specced survival. Needless to say I died a few times at the start of my trip. I found it much more difficult to kill multiple mobs as my pet seemed to lose health quickly or I pulled aggro. I think I would recommend a Beast Mastery spec for leveling. Your pet is much stronger and will increase your soloability.
These are just my observations so far. You experiences may have been different. I am just glad I am not solo leveling a Holy Priest......shudder.