For a few years, I used the Sennheiser HD 497 and have been very happy with them, with the sound as clear as a good pair of Hi-Fi speakers. These have been superseded since (looks like by the HD 457, 4 out of 5 rating on Amazon.com), but if you’re on a budget, these are very good for the price. I’m not a musician, so I can’t comment on how accurately they reproduce the audio.
For noise-cancelling headphones, unfortunately you generally get what you pay for and most of them are generally all over the ear type. They work by playing an inverse of the background noise to cancel most of it out. With most noise-cancelling headphones I’ve tried at an airport duty-free shop selling them, for the noise cancelling feature, they’re generally all quite effective. However, the problem with most of them is that the sound quality or bass suffers and the cheaper ones usually end up adding white background noise.
The main headphones I use now are the Audio-Technica QuietPoint ATH-ANC7. These are quite expensive (currently $179 on Amazon.com or Â£97 on Amazon.co.uk, 4.5 out of 5 rating on both), but I’m very happy with the sound quality of them. From what I’ve heard, they are the only ones that rival the Bose models. They take a single AAA battery, but the battery lasts a long time and I’m still on the original battery.
Just of curiosity, what type of noise would you like to block out? For lower frequency sounds such as while in a car, bus or especially on a plane, you’ll be far better off with a pair of noise-cancelling headphones than even a regular pair of the same price. To give an idea, the Audio-Technica pair I have brings the noise of a 737 (typical of Ryanair) down to that of regular headphones in a typical petrol driven car. For higher frequency background noise such as a birds tweeting, people chatting, etc., I don’t think it really matters as noise-cancelling headphones generally only soften these sounds and a regular pair with good sound insulation would have a similar effect.