I know for example the Concorde flew faster than sound, but I believe the options are much more limited now. Is it possible to be in a plane that breaks the sound barrier, without having access to flights in planes like the F16?
The only way that I know of (and I and @Stuart have both looked into this) is the MiG-29 flight. You need some training, but then there are a few options to do this with one (as a passenger - don't worry, you don't need to know how to fly it).
There are some sites on the net describing the options, but for example, flyajetfighter.com gives you several options, including:
3/ Stratospheric flight: reach the edge of space in the MiG-29. Fly up to 70,000 feet, twice as high as a normal airliner, and admire the curvature of the earth. The sky is pitch dark, and you are at twice the speed of sound. This is an extraordinary experience, and the only one so far to get you into the stratosphere.
Other than that, you might get 'lucky' in a regular passenger jet if something goes horribly wrong and you spiral into a spin and plummet towards the earth - but that's theoretical, and for obvious reasons, not something you want to experience.
There are other options (just see FlyAJetFighter.com and click the drop down list - but the MiG seems to be one of the more feasible options - there are several online companies showing this option with a brief google search.
Another example - the MiG-21 is available in South Africa with a few companies, the example I've linked to has a full adventure deal for around US$10,000.
Another site - includes many options and a price list, ranging from US$3900 to US$22990, depending on the plane, but it's not clear which of those will be supersonic flights.
And it's worth noting that most of these companies (all of them if I'm not mistaken) won't take you if you have any medical conditions - ruling me out :(