It really depends on what kind of solo music you want to play, in what environment, for what audience, and how you anticipate which trends to evolve in what way.
Just a few, in no particular order:
Historically Informed Performance: This has taken off a lot lately, and is coming out of the shadows of the "classical" music world. Dittersdorf and Vanhal and other solos using Viennese are being seen more and more on appropriate period instruments, or modern instruments tuned in 4 string Viennese tuning (without the low F) with or without tied on frets. If that's where you see your solo playing going, then maybe a Violone is a direction to go in.
Likewise, depending on the period you are focusing on, a three stringed instrument in 4ths or 5ths or GDG or solo 4ths BEA or some other tuning might be appropriate. Gut strings, a period bow or two, it is really up to you how far down that rabbit hole you want to go. These instruments do have limitations. You aren't going to be able to play a Violone in a modern bass section, three strings might limit some other possibilities etc. However, there is a growing and extremely loyal audience for early music in most big cities. If you break into that scene, you will see the same faces at EVERYTHING you ever do. It's kind of like a cult that way.
"Normal" 4 strings (in 4ths or solo 4ths): These are very standard, and most appropriate for a lot of settings. Some pieces of music and some composers, the Hindemith sonata is a great example, need this configuration. Extended technique, excessive use of harmonics, drone notes, double stops etc. are almost always written with this tuning in mind, so a lot of the bass repertoire out there becomes much more challenging when you deviate from this. Among the die hard bassist crowd, you meet some resistance when you deviate from this as well. Although there are a few out there, most "traditional" bass soloists like Gary Karr, Thomas Martin, Hal Robinson etc. play "normal" 4 string instruments.
5 strings with low B: Very rarely do you see solo music that takes advantage of this tuning. I haven't encountered any so far, although I am sure that some has been commissioned for this tuning. These instruments tend to be very large and getting up to the "solo" register isn't as easy because of how they are built. They often have a big pipe organ, orchestral section, "cannon" sound which is very appealing for that kind of work, but not necessarily what you are looking for for solo stuff. Also, they are challenging to find in the bass market, because they are considerably rarer than 4 string basses.
5 strings with high C or occasionally high D: We are seeing a lot more of these in non-standard/classical worlds. Jazz players, world music guys, genre bending type stuff like Renaud Garcia-Fons are really taking the bass to new heights with this. If you plan on thinking/playing outside the box, that is definitely a direction to check out. The instruments are still challenging to find much the same as a 5 with a low B. Again, there isn't a lot of repertoire specifically written for this tuning, but a lot of standard bass rep can be played in it a 4th down the fingerboard, which can be advantageous for the stuff that really gets up there.
As previously mentioned, some die hards aren't a fan of this. The top string "doesn't sound like a bass" is the most common argument/detractor, which in some circles is a backhanded compliment or insult.
5ths: Joel Quarrington, Paul Unger, Silvio Dalla Torre, Tomoya Aomori just to name a few. This can be done on "standard" instruments, or not so standard like the Bob Spear instruments. It sounds great, a lot of traditional double bass repertoire is still very possible, and it opens up some Cello/Violin repertoire as well because of the tuning similarities. It resonates really well and is great for carrying over an ensemble or piano.
Other options: very small upper bouts, shorter string lengths, fingerboard extensions that almost touch the bridge, more than 5 strings, and various creative things have all been done with varying levels of success and acceptance. The farther you stray from normal, the greater chance you have of alienating other bassists, who will be the majority of your audience. For every guy out there that things those 10 string electric basses with a low C# string are super cool, there are 20 guys who think they're the worst idea ever.