I'm a diehard altoholic, and I tend to jump from one to another multiple times throughout the day. Its not unusual to play a dozen toons in a day. The downside of this is I didn't get anywhere near all of them to 50, and those I did were part of dedicated teams that ran together once a week. So, I can't speak much to soloing the end game. I also only started dabbling in sets before the lights went out back in 2012. So, keep that in mind. Other people will have wildly different views.
My take on soloing each build is as follows:
Blasters - Can solo effectively but you have to put thought into your attack chain and don't neglect the controls in your secondary set. You'll die often, usually spectacularly, but you'll take a lot more of them out each time you go. My main Blaster was Ice/Fire.
Controllers - Cannot safely for most of the game. Can't solo quickly until they get the pet. It's possible to solo mission and take very little damage because mobs are all locked down. The trade off is you're basically stabbing bears with toothpicks for 30 levels. There are a few exception to this. Gravity and Illusion can deal more damage. Kinetic makes everything better. My main controller was Fire/Rad.
Corruptors - I never really got the hang of these. They deal less damage than blasters and have the same secondaries as controllers. They can be effective, but my experience was they really shined with a partner or two. What I say below about the Defender primary applies to the primary here as well. My main Corruptor was a Sonic/Kinetic.
Defenders - Being able to solo means choosing your primary set wisely. You want powers that affect the mobs, or yourself. Force Field, Sonic, Empathy, Pain Manip, Cold, Thermal are all bad options for solo because they are mainly for buffing teammates, especially at the lower levels. On a team, they rock, but solo, there's just not much of use. Radiation, Poison, Dark all debuff the enemy and that means you get a benefit. Kinetic buffs the WHOLE team, even if that's only you. I didn't have a main Defender. I had several. Dark Archery, Rad, Dark, Empathy Dark,
Dominators - I never could play them. Nothing I say about them would be of value.
Masterminds - Soloing a Mastermind can be done a few ways. You can be active, controlling which pet attacks each target, buffing and debuffing as you would any other team, or you can set them on aggressive and just walk while you watch a movie on the other screen. If you want to play one of the non-solo friendly defender primaries, do it on a Mastermind, because you'll always have a team. Again, I had several. Necro Poison, Mercs Traps, Robot Dark.
Sentinels - These didn't exist back then but from what I've seen on the ones I have so far, the early game is a snap. You have corruptor level damage output and nearly scrapper level armor/defense. What you don't have are soft controls, so you'll need that armor.
Stalkers - Another AT I never got the hang of. They are kind of like the melee version of blasters. They deal a lot of damage very quickly, but the longer the fight went on (at low levels) the less change I had to survive it.
Scrappers - Lots of melee damage, and pretty tough. It can take a while to get them slotted, because you want to flesh out the attacks and the defense, and there just aren't enough slots available. Highly solo playable. Main was a Dark Regen, but I also had a Dark Dark that I really liked.
Brutes - Better mitigation than scrappers, but not quite the same damage output on average. Brutes give you the option to work up your mitigation slotting first, and let Fury be your damage slotting until you're durable enough. Main was Fire Dark
Tanks - Safest AT you can play for solo. Damage output is low, and that can make combat feel like it's dragging on. Defenses come fast, and that makes slot distribution easier. Main was a Invlun Axe.