MP3 search engines seem to appear by the dozen every week and about 11 of them disappear after a while either because no one uses them or because they have been sued by a record company. I have seen options that help you against this coming and going of search engines. The first is to search and play only music from "legit" sources like Youtube or Last.fm, the other is to host the mp3 search engine in a country that is outside of the scope of organizations and rights holders.
MP3shki.ru does both. It crawls for music at websites like Youtube or Last.fm and is hosted in Russia. The interface is one of the best that I have seen yet, graphics and usability is awesome. Begin by clicking on the little EN link on top to switch to the English user interface, makes it easier.
Then search for an artist or song and watch how the search results are populated on the same page. A click on a search result opens a tiny overlay menu with options to play the song, add it to the playlist, to view the source and the link to the song and to mark it bad.
Songs can also be dragged and dropped into the playlist. It is furthermore possible to create more than one playlist, I suppose you can save them if you register an account.
Update: The Mp3 search engine website has been removed from the web and is now redirecting to another Russian website that unfortunately has nothing to do anymore with the original website.
Probably the best alternative in this regard is to hit YouTube right away to find your music fix on the site as it is very likely that you find the majority of songs that you are looking for on the video hosting site. Last.fm as mentioned earlier is an option, as are other great resources like Jamendo.
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