Yahoo! is not a search engine, even though it provides a search facility. I'll cut n paste the following which I found on their site (help section):
Your search results will contain matching web pages and web sites provided by third-party search engine providers and the Yahoo! Directory.
From the above, it is obvious that it relies on 3rd party search engines to do the job.
With regard to popularity, I would say 'Google'. People have even started using it as a verb, in place of the word 'search' :) Something like what 'Xerox' has become in the photocopying world.