I'm sure there are a lot of different opinions on this, but as far as I've seen there really isn't an "uber" class. All the classes can solo effectively, so what it really comes down to is what you enjoy playing.
Here's some of the things I've noticed.
Minstrel:
In groups: Expected to be a dedicated healer and not much else.
Solo: Does extremely well on 1 on 1 similar level mob fights in the standard stance. 3 or more of anything can be serious trouble.
What drives some people nuts: Solo encounters can be long and slow when compared to other classes.
Note: There is a stance that comes into play at later levels that allows the minstrel to have greatly increased damage output at the expense of healing abilities being reduced to half.
Guardian:
In groups: Tank
Solo: Is very gear dependent, but if appropriately geared at all times (standard best available for level, player crafted, non-single use, is generally fine) can stand toe to toe with just about anything under a regular red con extremely well.
What drives some people nuts: Solo fights are nearly minstrel long and are almost attrition wars. Damage output isn't much greater than the minstrel in the standard stance.
Note: There's a stance for this class at later levels that increases damage output, but forces you to use a 2 handed weapon thereby knocking about 1/3 off armour value.
There are probably a lot of details I'm missing on the ones below this as I haven't played most of them past 15 or not at all.
Lore-master:
In groups: Crowd control and power management This class is in VERY high demand for groups, especially instances.
Solo: Great solo capabilities, especially as the pets get better. Can be very "squishy" at low levels. Has loads of tricks (eventually) to handle multiple mob encounters.
What drives some people nuts: This class is often considered "too complex", which is also a common complaint of the crowd control type classes in most games.
Hunter:
In groups: Fast, big, but with an emphasis on self-restraint (big damage = bigger aggro generation), damage output. Traps come in very handy when a CC class isn't available. Group transportation certainly gets things moving faster.
Solo: Can generally take down an even con regular mob faster than any other class in the game, often before it reaches the hunter, especially when using traps. Has a movement speed buff, and farms mobs faster than any other class could even contemplate, so it's very useful for an avid crafter to have one in their toon line up eventually.
What drives some people nuts: This class is by far the most frequently played in the game, by a large margin, and is also the easiest class to find an example of it being poorly played in groups. Players seem to complain about hunters a lot.
Note: There is a stance that helps with aggro management at higher levels.
Captain:
In groups: The buffer! Has a great on-defeat of a mob group heal that is a huge asset as well. Has been known to tank in a pinch.
Solo: I never played one far enough to know, but then I've never heard of one complaining that they can't do it effectively.
What drives some people nuts: I've heard the complex complaint a lot for this class, although I'm not sure how valid it is.
Note: This is a very high demand class for groups and there just never seem to be enough of them to go around.
Champion:
In groups: Damage output, reserve/secondary tank. Can tank if necessary.
Solo: Has lots of multiple mob strikes that make similar level multiple mob encounters less of a hassle.
What drives people nuts: It's the jack-of-all trades fighter and is sometimes accused of being too straight-forward. (It's also probably why Turbine claims it to be the perfect first character class, especially if one is not a seasoned MMO player.)
Note: In my opinion, this class paired with just about any other makes a very solid duo.
Burglar:
In groups: CC, burst damage, conjunction triggering, scouting around.
Solo: This class can get to a lot of places that almost any other class would have to clear a lot of mobs to get to, which is more handy than it may seem on the surface. The class is a little "squishy" but has a lot of tricks at its disposal.
Note: This is also a high demand class that there seems to be a general shortage of in the game.
Somebody else could comment better on this one especially. I genuinely did find this class to be pretty complex and I just wasn't getting the hang of it.
I'd say pick a few. Play those to 15 or 20, which doesn't take all that long once you do it once. Then go with the one you like best.
Fair warning though, some classes really do take flight at higher levels. For example, 36 is a huge one for the Lore-master.