The company OCLC (seller of library services) has published a full report on a survey they commissioned from Harris Poll Online (a market research company) about perceptions of libraries and competing products and services (in particular bookstores and search engines). The 290 page report is available in full or section by section from: http://www.oclc.org/reports/2005perceptions.htm
There were 3348 respondents, 450-500 each from Canada, UK and "Australia, India and Singapore" (latter treated as one region), and the rest (i.e. over half) from the USA. The researchers claim a hugh confidence level in terms of generalisability to the internet-using populations of the respective countries, though I would want to know more about how they selected the sample and how representative Harris' database of online users was to start with. They don't compare their findings with anything much else except some general stats of what % of the population uses the internet (i.e. they don't compare their findings with other surveys about people's use and perceptions of search engines, or stats on library use in the respective countries etc.), and knowing something about UK stats makes me think that you can't generalise these findings to the whole UK population.
However, having said all that, there's some interesting stuff in there. There are questions about perceived relative trustworthiness and use of search engines & libraries, extent to which people have used/heard of search engines, libraries and bookstores, positives and negatives about libraries, and so forth. A good sized sample of free-text responses to questions like "what's the first thing you think of about libraries" is given (by the way, the answer is that most people think - "books"). The conclusion (related to this "book" dominance in perception of the library brand) is that "It is time to rejuvinate the library brand" (section 6-8) Hmmm.
de Rosa, C. et al. (2005) Perceptions of libraries and information resources: a report to the OCLC membership. Dublin, Ohio: OCLC. http://www.oclc.org/reports/2005perceptions.htm
(Photo by S. Webber: Rose Apple tree (flower), Sydney Botanic Gardens, Australia, Nov. 2005.)