In a previous post I discussed my concerns with Google's search indexing, which caused me to make the switch to Bing. While I still like Bing from a user perspective, I have found anecdotally with my own website that it is very slow to index new content, despite my submission of my site map file to their platform.
Recently I became aware of DuckDuckGo and decided to give them a go for a few weeks as my primary search engine, and have been very happy with the results. They also have a strict moral stance on user privacy which appears to be unique amongst search engine providers, which you can learn more about here in this video interview with the founder of DuckDuckGo, Gabriel Weinberg:
You can find out a lot more about DuckDuckGo on their about page: About DuckDuckGo
Updated 2022 : note that the above post was originally published in 2011, but is left here for archival purposes.