Hello there,
One of the drawbacks to a series capacitor solution vs transformer is there is no galvanic isolation with a single capacitor. Even with two capacitors it will not achieve this end goal. Some electronics forums even go so far as to ban discussion of these circuits which are collectively called, "transformerless power supplies", or, "offline power supplies".
With a regular two winding transformer that passes the UL test we get two forms of isolation:
1. The two windings are not electrically in contact with each other.
2. The two windings are not physically in contact with each other.
#1 just means that there is no current path between windings so chance of shock from the line itself is reduced.
#2 means that the two windings are not wound one on top of the other as in industrial circuits, but are actually separated by a distance along the core. That means they cant touch even if the insulation breaks down, and also the arc-over distance is very large.
So these two combined make for a very safe power supply, and that is required in almost all wall warts made today.
Now with a single capacitor you dont have this kind of isolation because for one thing one side is always connected to the line. If that one wire happens to be the neutral wire, you might be ok, but if the plug is pulled out and put in in reverse, you end up with the full line voltage on that one line which can cause a HOT to GROUND shock to the user.
The other wire isnt that much better anyway though, because the capacitor has the ability to pass a significant current level for a short time which is also dangerous, but even more to the point is the cap can short out and that is a typical failure mode for caps, so you get shocked that way too.
The other view to the cap power supply is that it is used in a lot of professionally designed equipment, so how can this happen? It just so happens that when this technique is used, the equipment must provide other means to protect the user. For example, plastic shafts on potentiometers that prevent any direct contact to the internal circuit.
The usual recommendation is that beginners to electronics should not use these cap circuits. That is probably because there is a certain expectation of safety associated with any voltage level. For example, if we hear, "it has a 12v output", then we think it is probably safe to use as any other power supply. In fact, a cap power supply that puts out 12vdc may in fact also be able to put out 120vac with respect to earth ground, so it becomes a safety hazard.
So in the end if you dont realize the danger then dont use it. If you do, then be careful.
The rule of thumb is keep one hand in your pocket while making voltage measurements. In other words, never touch the live circuit with both hands at the same time. Also make sure your feet are not grounded :-)
Good luck to you.