I've played 4 chars:
Human Paladin - best solo, easiest to play, least frustrating, but less thought required. Verigan's Fist is outrageously powerful and will make you unbelievably incredible until the low 30's. That's my assumption at least, being level 25 with it and still being very overpowered. In general, 6-12 seconds of invulnerability and super high armor mean you don't die too often. And though the hunter CAN have minimal downtimes, at some level ranges the hunter has a lot of downtime due to low mana. The paladin generally will have less.
Night Elf Hunter - second best solo, but until you get feign death, you will tend to get into some sticky situations that are tough to get out of. Feign death really makes the hunter a LOT of fun for me. The hunter is otherwise a lot of fun, probably 2nd most fun overall. The pet will save your life about as often as the paladin's invulnerability, but the low armor is the main cause of sticky situations until feign death. I found that being in a group as a hunter can be frustrating at times, especially in instances where space is limited and you have to find the right distance without pulling nearby enemies. It also is very frustrating when the looting is set to group loot and you end up never getting enough quest items before the melee people leave the group.
Undead Rogue - very interesting and fun, but I haven't played it quite enough to know how things will go once the "core" skills are obtained. In the early levels (only thing I've done), the rogue does quite well. In groups I've been in with rogues, I've really liked what they brought to the table. If you have a rogue and mage, your group is going to have a firm grip on crowd control. Like the mage, though, it is frustrating (even as a hunter/paladin) when other group members screw up your debilitating skills (sap, or poly for the mage).
Gnome Mage - this is by far the most challenging class until you get to level 10, when frost nova arrives to save you from the umpteen deaths per hour. I'm imagining that blink will be similar, haven't quite gotten there yet. I like the mage a lot, it seems very challenging and requires quite a bit of planning and pacing. Conjured food and water is probably the most appealing aspect of the mage so far.
I'd say the paladin is the best class for soloing (haven't tried the shaman), with the hunter second. The mage is the hardest early, but since I haven't played the rogue or mage too far yet, I can't say whether that changes later. I do suspect that the rogue becomes quite good once sap and vanish comes into play.
As far as most fun, mage by far for me. But mostly because I like to think, plan, and pace my battles.