As of now we have some broken trinkets (such as
https://www.wowhead.com/item=178769/infinitely-divisible-ooze) that help immensely in soloing so all classes can do it.
That said if it ever get fixed some will have trouble (mostly Rogues and to a lesser extent Mages). If you are a Veteran and have
https://www.wowhead.com/item=173125/revitalizing-jewel-doublet Mages will still be fairly viable but even with these Rogue struggle.
If I had to give the best classes for soloing I'd say Resto Shaman because it's broken, Guardian Druid because of the decent damage and
https://www.wowhead.com/spell=5215/prowl or Prot Warrior because it is unkillable. Hunters (Surv and MM) and Warlocks (Destro and Affli) are excellent with their pets but some bosses require to manage threat.
That said Brewmaster Monk and Protection Paladin are perhaps the easiest classes to start with if you aren't used to soloing. Monks have an excellent well-rounded toolkit and Prot Pallies are also unkillable with one of the best toolkits at out level. Priest is also very easy to play : Discipline is perhaps the most unkillable of all specs at level 20 at least in my opinion and Holy, while a bit squishier, has some very good damage.
I haven't played a lot of Evokers but they seem decent to me.
Classes like Demon Hunter or Death Knight are a little bit tougher to make work but they are plenty viable.
So yeah Shaman is the best class at level 20 but Paladin/Monk are a bit easier for beginners.
I'm currently writing some class guides so if you want a more in-depth analysis it's a work in progress.
Only have DKs so far so I'll link it here just in case it can help you :
https://xpoff.com/threads/how-to-solo-a-death-knight-dungeonning-guide.100407/
Hope that helped ^^