Well, I've done a lot of solo diving over the years, and the gear varies.
In the US Air Force, we soloed each time we went on a parascuba jump. This was the time period of 1967 to 1977. We jumped with a set of double 42s (steel converted life raft CO2 bottles, rated at 2100 psi), a double manifold with a J-valve, and a single-hose regulator which was just that, a single hose. I have many water jumps in this configuration, without a mishap. We at times needed to descend to about 90 feet for parachute retrieval, etc.
Later, when I was going to school, I dove the twin 42s with a double hose regulator on a number of scientific dives, some of which were solo (due to water visibility). Again, sometimes I wore either a double hose regulator or a single hose reg.
In 1973, I dove the Warm Mineral Springs Underwater Archeological Project, where they introduced me to the concept of an octopus (I was a NAUI Instructor at the time, and octopus regulators were not yet common). I have been diving them ever since. I bought a Trieste II double hose regulator about that time, and set it up with an octopus. I dove it both buddy diving and solo mainly with my twin 42s. I also dove single 72s with double hose regs (Dacor R4, DA Aquamaster and Mistral), and with some single hose too (AMF Voit MR-12).
After marrying and settling down in a small community without many divers, but with a nice river, I began diving solo in earnest. The river was great fun for diving, and I learned a lot. I also dove many combinations of gear. The twin 42s had to go, as they were deamed unsafe by the USAF (and my contacts in the Air Force told me they had drilled theirs, so I drilled mine--I was really sorry to see them go though). I bought a set of twin 42s (Sherwood) with a Sherwood double manifold, and began a lot of experiments on the double hose regulators. My favorite was a Healthways Scuba Delux, which I put USD hoses on and used with a single hose for a backup--sometimes. I felt (and still feel) very comfortable diving to 35 feet with only one regulator, no octopus, and no backup. If things go wrong, surface! It's that simple.
I dove singles too, and enjoyed the Scubapro AIR I/Mark VII combination with an octopus for many years (and still do). I don't think there is a regulator made that can out-perform it yet, especially on exhaust exhalation resistance. It is a wet breather though, especially in certain positions.
I developed various concepts in buoyancy control during this period, and finally patented a front-mount BC with pockets for an octopus second stage in about 1986. I still use it today--the most expensive BC ever built, as only two were ever made, and the patent cost me a bunch of $s.
My thoughts on solo is that you need to be very comfortable in the water first. That means comfortable with all your gear, comfortable that you can handle situations where gear malfunctions (regulators leak water, fins are sometimes lost in current, etc.), and if you are that comfortable, then solo is a very neat way to dive. If you are shallow, and have the proper skills, you don't need a lot of equipment (like Nemrod says above).
But I draw a definate line for safety too. That is two atmospheres of diving solo. I won't dive greater than about 60 feet solo. If you are deep, dive decompression, dive inside a shipwreck or under an overhead obstruction, you need a COMPETANT buddy. I also happen to believe that if you dive a decompression dive, you need a recompression chamber on-site, immediately available, to do so safely. To dive DIR, without a decompression chamber available and on decompression dives, you are inviting a book to be written about your dive, such as was written about in the book The Last Dive. If conditions don't look right, if I don't feel right, if something is out-of-place, on solo dives I simple either don't dive, or abort. There were several times last year where I got my gear together, drove to the river dive site, looked it over, and decided to go to the bicycle shop instead. But there were several other times where I dove, and had a great time.
Concerning the usefullness of an octopus second stage when diving solo, there are a couple of times they come in handy (but it is very rare).
--I was diving at Edmonds Underwater Park when my Trieste II double hose regulator suddenly would not give me any air; I switched to my octopus and kept diving (I modify my regulators, and a venturi baffle I had glued into the mouthpiece had come loose, and blocked the non-return).
--Last summer, my DX Overpressure double hose regulator developed a hole in the mouthpiece, which gave me a good spray of water whenever I inhaled. I was wearing my twin 42s, with a backup Sherwood Blizzard regulator, which I switched to.
I guess these examples were all with older-style, modified gear, but they do show that there can be a reason for a solo diver to wear an octopus. But again, the solo diver could also just surface too.
SeaRat