Let's pay some attention to recording dates here: the Furtwangler Ring operas were recorded in the early 1950s. The Solti/Vienna Philharmonic Ring was recorded 1958-1965. The Bohm/Bayreuth Ring was recorded 1966-67. The von Karajan/Berlin Philharmonic Ring was recorded 1966-1970. The Boulez/Bayreuth Ring was recorded 1979-80. Why is this important, and why, you might well ask, aren't there more recent recordings of the Ring that are serious contenders? Because the sad fact is that the 1950s and 1960s were the last decades in which it was possible to put together casts of singers adequate to the extraordinary vocal demands of these operas. Roles like Siegfried, Brunnhilde, Wotan require heroic (helden) voices that not only need to be huge and powerful but need to have great stamina (these operas are very long); such voices have always been rare and today are unfortunately nonexistent. (Those who really know and love Wagner will tell you that you have to go back to Melchior for a great Siegfried, Schorr for a great Wotan, and Leider, Flagstad, Traubel, or Nilsson for a great Brunnhilde.)
With one exception (a 1954 EMI Die Walkure with the Vienna Philharmonic which was his last studio recording), the Furtwangler Ring operas have Italian orchestras and choruses, and it has usually been said of these performances that they are disadvantaged by rather dim monaural sound and by the inadequacies of the Italian orchestras and choruses he was working with in delivering polished, first-class, idiomatic Wagner performances. Beyond that I'm not going to comment on them because I don't know them. They have their adherents.
Of the Solti, Bohm, von Karajan, and Boulez Rings, I think the Solti/Vienna Philarmonic set (a studio recording) is the clear first choice. If you consider the quality of the singers overall (with the last great Brunnhilde in Nilsson), the quality of the conducting and the orchestral playing, and the quality of the recorded sound, this set is really in a class by itself. It has long been praised as a true classic in the history of recording, and it deserves this fame.
The Bohm/Bayreuth Ring would be my second choice. These are live performances from Bayreuth; the singing is generally in capable hands (Nilsson is again the Brunnhilde); the sound is generally good (definitely not the in class of the Solti set), although there are inevitably some awkward moments where balances aren't quite right, you can sometimes hear the prompter, singers are sometimes too close or too distant, etc. The Bayreuth orchestra is good and the famous Bayreuth chorus is excellent. The main problem with this set lies in the limitations of Bohm as a Wagner conductor: he is simply not in the class of Furtwangler, Solti, or von Karajan; again and again his pedestrian conducting fails to make the most of opportunities and to realize the dramatic possibilities of this scene or that moment; this is a serious, pervasive shortcoming here.
The von Karajan/Berlin Philharmonic set (studio recordings) would be my third choice. There is a high standard of conducting and orchestral execution here, and the recorded sound is good (although, again, certainly not in the class of the Solti set). The main problem here is in the unevenness of the singers and some odd, questionable casting choices; this Ring has been criticized as an artificial studio production using some singers who wouldn't really be up to the demands of their roles in the opera house.
The Boulez/Bayreuth Ring would not be a choice of mine at all. It is disfigured by Boulez's revisionist views of Wagner, with results that are often ineffective, and pretty consistently undistinguished singing. Clearly by the time these recordings were made, even Bayreuth, that mecca for Wagnerites, could no longer put together an adequate cast for the Ring. Again, the Bayreuth chorus shines.
All things considered, here's my recommendation: get the Solti/Vienna Philarmonic complete Ring (it's budget-priced as a complete set), and then supplement this with the well-filled 2-CD budget-priced compilation of highlights from the Bohm/Bayreuth Ring and the von Karajan/Berlin Philharmonic Ring. (Don't get any of the 1-CD highlights discs; these are just too skimpy to give any real idea of the Ring.) This way you can have the best overall Ring and sample two of the worthwhile alternate versions. (Unfortunately there is no 2-CD highlights version of the Solti set.)