I’m going to go against the grain here a bit and not argue, as some people have done, that the Beatles were not all that as a band. But I do take a slightly contrarian stance. Here’s my thesis (aka, the TL;DR version):
John and Paul were first-rate singers and songwriters, and that was all that was really needed for them to succeed. Paul was/is a great bass player. John, George, and Ringo are/were good but not masterful musicians, but their idiosyncrasies actually were beneficial to the Beatles’ early sound. After the band became just a studio band around 1966, the musicianship became fairly irrelevant.
OK, a few notes on the individuals (descending in order of overall talent):
Paul. Since I have mostly only praise for him, I doubt there will be much controversy. He’s the only one of the Beatles who was/is truly a grandmaster on his main instrument, the bass. He also plays other instruments very well. Whether he or John was the better vocalist is simply a matter of opinion. I give Paul the edge. Paul’s songwriting is both a strength and a weakness. At his best, he is one of the greats, but he also wrote a lot crap that started to drag down later Beatles albums, and his solo stuff doesn’t do a lot for me. I like a small handful of songs.
John. Great vocalist and songwriter. I’ve never heard it argued that he was particularly good at any instrument. It’s no coincidence he mostly played rhythm guitar. John’s taste level in his solo career was better than Paul’s but a lot of it is pretty dry to me. Again, I like a small handful of songs.
George. I rather like George’s singing while with the Beatles, and I am fan of several of his songs (including “Don’t Bother Me,” which tends not to get noticed). I also think that he was an idiosyncratic guitarist whose playing was essential to the Beatles’ sound when they were, you know, a band that did concerts and stuff. In that sense, he is certainly worthy to go down in pop music history as an important guitarist. But I don’t think he was technically great, and I’ve never heard any evidence in the form of music to give me that impression. Further, he did not do the famous solo on “While My Guitar Gently Weeps”; that was by Eric Clapton, who was uncredited on the White Album. That just blows my mind. Both things. The fact that George would not do the solo on his own song, and the fact that Clapton was not given credit, which strikes me as dishonest, as though everyone involved was fine with having people think it was George’s work. As for George’s solo material, it has never grabbed me, which surprises me, as I think his songwriting for the Beatles proper was rather innovative, if not always completely successful. George deserves a slap on the wrist in heaven for the awful 80s cover “I’ve Got My Mind Set on You.” Horrible.
Ringo. No one argues that Ringo is technically a great drummer, and he freely admits that himself. But here again his idiosyncratic sound (partially due to playing left-handed on a right-handed kit) was an asset to the Beatles when they played live. I also like Ringo’s vocals, and the Beatles utilized them effectively. As for solo material, I probably like Ringo’s stuff best of all, as he has only a few songs I like, but I am more enthusiastic about them than Paul’s or John’s. I would say “It Don’t Come Easy,” which he wrote himself, is better than any solo song by the other three. Ringo was probably the least talented of the Beatles overall, but he certainly leveraged what he had very effectively.
By the way, when I say “idiosyncratic” above, I don’t mean that the playing was weird. What I mean is that the playing had unique qualities based on limitations that make the sound unique and good. While George’s playing wasn’t masterful in the way the playing of a session musician who can just play anything on the spot is, it’s immediately identifiable. Early Beatles songs (contrary to many, I like the period up to and including “Help!” the best) would not be as good with blazingly skilled guitar solos.
OK, that’s my take. Thoughts?