As your question implies, there is not really a "best" RAID configuration, only best for a particular set of circumstances, with cost often being one of the most important factors.
Without going into the minutiae of controllers and software, here would be my rules of thumb.
RAID 0 is the fastest since you can read from and write to many disks at once, and no space is "wasted" for redundancy. Lose any disk and you lose the set, so RAID 0 should only be used on a machine you do not care about, or is easy to restore and does not contain data you value. A gaming machine might fit this scenario, though honestly, the speed difference is not so noticeable that I would be willing to accept the increased risk of having to rebuild the machine. It can also be useful for really fast "scratch" areas if you have software that needs that sort of thing.
RAID 1 is the very common mirror setup, usually of 2 disks. Reads are generally just shy of twice as fast as a single disk, while writes are just a bit slower than a single disk. On servers, RAID 1 is an excellent choice for your operating system files. It is also a good choice when you need redundancy, your storage needs are not so great as to require RAID 5 and you might benefit from the extra read speed (database log files are commonly placed on RAID 1).
RAID 2 through 4 (not in the question) are generally unused except by certain vendors' products on the the enterprise side.
RAID 5 is a compromise between not wasting too much space on redundancy and still getting the added performance of extra disks. Read speeds are very good since all the disks can participate. Write speeds can sometimes be a problem with RAID 5, though I think this is sometimes overstated depending on the situation. Even with hardware that does the parity calculations, small random writes suffer on RAID 5 since each logic write operation requires 4 I/Os (read from data disk, read from parity disk, write data disk, write parity disk). Pick RAID 5 when you want to maximize the amount of storage you get out of a disk set while still having a good degree of safety. Avoid when your application has high performance needs and requires lots of small random writes (virtual machine hard disks, database data files). Also be aware that modern large disks take a loooooong time to rebuild when they fail as part of a RAID 5 set, and that puts your data at risk longer for a second disk failure. RAID 6 can reduce this risk at the expense of even worse random write performance.
RAID 1/0 and 0/1 have pretty much the same performance characteristics. 1/0 is preferable though because a failure means rebuilding only the pair of disks involved rather than an entire stripe of disks. RAID 1/0 is the fastest general purpose configuration of the ones mentioned in the question. Read and write performance are both great (reads can happen from all disks at once essentially, writes have to happen to multiple disks but no read/read/write/write cycle needed like RAID 5) though RAID 5 can win in certain circumstances. RAID 1/0 is (unsurprisingly) also the most expensive in nearly all cases. It should only be used when performance is critical, the data is highly valuable, and the application has no tolerance for down time. Often database servers (or at least their data files) make good candidates for RAID 1/0. I also prefer to put virtual machines on RAID 1/0 if there are going to be a lot of them or they run disk intensive applications that I really care about.
Solid State Drives can indeed be put into RAID configurations, particularly for its redundancy value. Many SSDs can outperform the bus (SATA) they are attached to though, especially for read operations, so the performance effect of RAID is less compelling with SSDs.