Well, since you asked...
The output voltage of most wall-warts (those ubiquitous black plug-in power supplies) is unregulated and poorly filtered. For example, a device marked "9VDC, 0.25A" will make 11 or 12 volts, pretty clean, with no load, and around 9 volts, but with noticeable ripple, at full load. Not great, but it will have almost zero effect on the output of most Zapppers. Why? Because Signetics designed the 555 timer circuit, the heart of most Zappers, to reject power supply variations. So while the Zapper output voltage level will vary proportionally with the power line voltage, the output frequency (and harmonic content) will remain remarkably stable.
Beyond portability, even a cheap AC supply is superior to most battery systems. Besides amplitude stability (no output droop as the battery voltage fades), the output impedance also does not change with time. ALL battery chemistries dictate that the output impedance of the battery increases as the battery discharges. This leads to a small frequency drift and larger output voltage droop.
If you want to buy one rather than use one laying around, get one of the adjustable or universal ones from Radio Shack. These have a regulated output; more stable and less ripple. Better.
Can the transformer "type" (construction?) affect the circuit output "quality"? Yes, but only if the designer is a bozo.
Finally - safety. The US consumes 100 million wall warts per year. Yup, one new one for every 3 people in the country, every single year. The most conservative estimate I've seen says there are a billion in operation at any given instant. Even if 1000 failed in a life-threatening manner (and they don't), that's million-to-one odds. Thanks to the steady advance of European safety agency rules, there is almost no difference between medical and non-medical small transformers. The difference between the two types of complete power supplies relates more to ground current (microamps vs milliamps) than to breakdown voltage or creepage distance. These are covered by UL 544 and 1950.
Who am I? Electrical Engineer, analog circuit and power supply designer for 30+ years.