Searching for status updates is not Twitter's forte, so leave it to Google to make its own Realtime Search engine more powerful instead.
Realtime Search combs Twitter and Facebook for public status updates. Google introduced the tool in December, sticking it in a sidebar under the category "Updates," but Realtime Search is now rolling out to google.com/realtime as well. (For now, you can experiment with the new features here.)
The most important new feature of Realtime Search, I think, is geographic filtering of status updates. By whittling down Twitter posts to your city or Zip code, you can find out, for example, whether there are any upcoming concerts worth seeing. Apparently, an 11-piece Kenyan band is playing a free show tonight, a few miles from my apartment. I'm not sure why Twitter, which turned on location-sharing in March, doesn't have a similar feature. (Click Image Below to Enlarge)
If you want these kinds of updates delivered to you, without pro-actively searching for them, Google now allows you receive status updates through Google Alerts, by selecting "Updates" from the "Type" drop-down menu. Either 10 or 50 results can be sent once per day, once per week or as they happen. This could be useful for a search term like "free food," which might not come up regularly in your area, but would be worth knowing about when it happens. Just one snag: You can't filter these results by geography, so your best bet for local results is to add the name of your area as a search term.
The last addition to Realtime Search is conversation view, which lets you see all the replies to a single status update. I like the intent here, but this is the one feature that Twitter should add to its own website. Some third-party programs already do this -- Tweetdeck, for instance, can display all replies as one chronological thread -- but trying to read a conversation on Twitter itself is like falling down a rabbit hole.
Update: It's been pointed out that search.twitter.com allows the robust search features described above, under "advanced search." This search feature is not available from Twitter's main website -- you have to know about it by name -- but it's there.
I don't think Google's trying to push Realtime Search into the mainstream with these changes. If that was the intent, status updates would be moved up in regular search results, not moved out to their own URL. Realtime Search is a tool for power users. Keep it in mind next time you're looking for the nearest taco truck.