The outlet in the built in hairdryer in the bathroom is clearly labelled shaver only. Plug anything more powerful in there and you risk poping the safety breaker.
As for the other outlets they are 3 prong, but, the way the plug is mounted close to the top of the desk, you need some kind of extension cord if the plug is any bulkier then the kind found on the end of a lamp cord. Typically a lot of hair dryers/curling irons have a big short circuit safety plug that will not plug in there.