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Silverstein et al., Analysis of a Very Large AltaVista Query Log, SRC Technical Note '98 looked at 43 days of queries in 1998 and pulled out how many terms there are per query, what the top 25 queries were, and how correlated query terms were.

There are various papers on an Excite dataset from the 90s, which include similar analyses to Silverstein et al., plus additional results such as the distribution of searches into different categories and the distribution of term frequency in queries.

Ozmutlu et al., A day in the life of web searching: an exploratory study, IP&M '04 looked at 10007 queries on a Norwegian search engine (alltheweb.com) and found substantially different search characteristics than reported in other studies. However, AFAICT, this study wasn't repeated to determine whether or not the differences stemmed from some difference in the search engine or differences in Norwegian searches. For example, this study found many more terms per query than the Altavista and Excite studies. Studies on "classical" IR systems also find many more terms per query than we see on web search. It's possible that the differences stem from a difference in how alltheweb.com is used as opposed to some difference in how Norwegian searchers assemble queries. That could probably be found by looking at Norwegian vs. other queries on a single search engine like Google or Bing, but I haven't seen a study that does that.

Park et al., L&IS '05 looked at search queries on a Korean search engine, NAVER. As with Ozmutlu et al., it seems difficult to determine if any differences from U.S. search results are because of quirks in the search engine (Taghavi et al. observe large differences between search engines even in the same locale) or because of a geographic difference.

Taghavi et al., An analysis of web proxy logs with query distribution pattern approach for search engines, CS&I '12 looked at queries that went through the squid web proxy over 9 months in 2010 and 2011. They presented how many terms there were per query, how that varied by search engine, and how that varied over time over the 9 month period.

Some things that are notably missing are:

  1. How queries have changed over long periods of time.
  2. The distribution of terms in queries. For standard search engines using posting lists, or even non-standard search engines using signature files, performance is critically related to the document-frequency of each term in the query. Analysis of the Excite data set shows something similar -- it was found that the frequency distribution of query terms was somewhat close to a Zipf distribution, but the frequency of terms in queries wasn't related back to the frequency of terms in documents AFAICT.
  3. How queries differ geographically.

Have any of these been studied and published or posted anywhere publicly or are there public query logs that would let someone else do this analysis?

There's some work on specialized sources of data like pubmed queries because query logs are (or at least used to be) available online, but the analyses I've seen are specialized enough that I'd expect the results to be quite different from general web search results.

A number of papers contain benchmarks that refer to query logs from various sources, but none of the ones I've seen appear to be publicly available. For example, Zhong et al., Optimizing data popularity conscious bloom filters, PODC '08 has a benchmark which uses "a partial query log at the Ask Jeeves search engine (www.ask.com) over the week of 01/06/2002–01/12/2002", but that query log doesn't appear to be publicly available.

In 2012, Wikimedia released a dump of user queries but they removed it almost immediately, with a note stating:

It appeared that a small percentage of queries contained information unintentionally inserted by users. For example, some users may have pasted unintended information from their clipboards into the search box, causing the information to be displayed in the datasets. This prompted us to withdraw the files. We are looking into the feasibility of publishing search logs at an aggregated level, but, until further notice, we do not plan on publishing this data in the near future.

As far as I can tell, there hasn't been an update on the wikipedia query log and there's no easy way to get a copy of the query log they posted.

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