The purpose of a blog search engine is to index blogs and to show some information easy to find in a feed, like the date of a post, the author or the tags associated with a post. Unlike web search engines, blog search engines have less sites to crawl and the feeds save them a lot of time, so they pick new posts much faster.
You should use a blog search engine when something important has just happened and you want to find out what people (and not the media) think about it. You can also use them for news that aren't in mainstream and for niche subjects.
I like Technorati's homepage because I can find what's popular right now: the top searches, tags, the most popular content. I like them because they show numbers next to each blog, like number of blogs that link there or the rank of a blog. Their numbers are simply irrelevant because the links aren't equal in importance, but it's nice to have the comfort of a ranking (it's similar to Alexa). Technorati is often down or pretty slow, but they have cool widgets to add to your blog or feed so I forgive them.
Google Blog Search is all about search. Unlike Technorati, Google Blog Search is not down or slow and sorts the search results by relevance. That may seem weird, so you can also sort them by date. Google's search engine is cool because you can easily change the time interval and create mail alerts for your favorite topics. For some queries, you'll also find a lot of spam, mostly from Blogspot.
Ask Blogs & Feeds wants to impress you. Blogs are a domain where Ask.com is strong (they own Bloglines), so their blog search engine must be good, right? Yes, but there are too many options. You can search for posts, feeds and news. You can sort the results by relevance, popularity and time. For each search result, you have the option to subscribe to the feed, send the content to your blog and to preview it right inside the search results page. That's cool, so why should I visit the blog? Ask's strongest point is the preview option and the option to sort posts and feeds by popularity, which uses Bloglines stats.
Sphere is good for only one thing: to find related posts and blogs. So you read something interesting and want some context. Sphere proudly shows you something to read. You can add the "Sphere it" button in your browser and use it when you get lost in a strange article.
Icerocket is a mix of Technorati, Ask and Google Blog Search (in fact, they copied a lot from Google's interface). It's also the best-looking blog search engine, but there's no way to sort the results other than by date or to restrict to a language.
The most original blog search engine is Findory that personalizes the results by analyzing your clicks. It's far less comprehensive than the others, but you'll love the search results recommended just for you. It's a good idea to create an account and to add your favorite blogs.
So if you want to have some idea about a feed and its popularity, try Technorati. If you want good results and a minimalistic interface, go to Google Blog Search. To preview search results, use Ask, and to find related posts, Sphere could help you. Icerocket tries to be a bit of everything, but it only partially succeeds. Findory shows mostly popular blogs, but it's a good filter if you don't want to read too many posts.
It would be nice to have a search engine that sorts the posts by date, but it also takes care to remove irrelevant results (including spam). A blog search engine that clusters the results by topic, like Google News, but also shows the connection between posts and their dependency. A way to monitor what interests you and to filter the good stuff.