Here's a compilation of "ask a librarian" services in 12 of 50 states, and as many countries as I could find. Whenever possible, these will have a (verified) note by them, signifying that I was able to ask and receive an answer to a reference question.
This is a work in progress.
Library of Congress Ask A Librarian (Library of Congress). Organized by reading room; several have chat based help available, others are via email or phone. "The primary mission of the Library of Congress is to serve Members of the Congress and thereafter, the needs of the government, other libraries, and members of the public. The Library's staff will respond to reference and information requests in accordance with this mission."
Alabama. Mobile Public Library Ask A Librarian. Call (251) 208-7078, or fill out an online form. No chat.
Alabama. University of Southern Alabama Library Help. Call (251) 460-7025, fill out online form, or via chat.
Alaska. Consortium Library Ask A Librarian. Call (907) 786-1848, or reference via email or chat. Consortium supports the University of Alaska at Anchorage and Alaska Pacific University.
Arizona. University of Arizona Ask a Librarian. Call (520) 621-6406 for the main library information desk, or reference via email and chat.
Arizona. Arizona State Museum Ask a Librarian. Reference via email, including research assistance related to archaeology, ethnology, and anthropology; questions answered in 3 days.
Arizona. Scottsdale Public Library Ask a Librarian. Reference via Questionpoint, a forms interface from OCLC.
Arkansas. University of Arkansas Ask a Librarian. Call 479-575-6645. 24x7 chat available; also reference by email.
Colorado. Boulder Public Library, Contact Us. Reference: +1 303 441 3194.
Connecticut. Darien Library, Contact Us. Information services: (203) 669-5236. Chat available during business hours.
Delaware. Ask a Librarian Delaware. Statewide service through Questionpoint.
Florida. @AALFlorida (Twitter). "We provide Florida with reference via live chat or texting FOR FREE from 10am-midnight Sun-Thurs ET, & 10am-5pm Fri & Sat ET. Email us 24/7."
Georgia. Georgia Tech Library Ask a Librarian. Call +1 404 894 4530 or +1 888 225 7804, or reference by chat, email, or text.
Michigan. University of Michigan Ask A Librarian. "Ask a Librarian is tailored to the research needs of UM students, staff and faculty, but any one can inquire about specific UM library resources and services. You can connect to a librarian by IM, email, phone or in person, and now, via text message." Call [+1-734-764-9373] for general reference and reference referrals, or text to +1 734 531 9275.
Michigan. Michigan State University Ask A Librarian. Call 1.800.500.1554 or 517.353.8700. A Reference Librarian is available by phone during hours when the Main Library Reference Desk is open. Reference also available by text message and by chat 24/7.
Washington. Ask a Librarian - Washington State Library. "We are here to answer your questions about Washington state government, history, culture, the federal government, and genealogy. We have an extensive collection of Washington newspapers, state publications, federal publications, and other sources for information." Answers delivered by email; obituaries may take as many as 6 weeks.
Thanks to Julie Weatherbee, Michelle G, Lou Rosenfeld, Eli Neiburger, John Blyberg, Peter Morville, Jessamyn West, Billy Barron, Sam, Caleb and many others for help in putting this list together.
Telephone numbers will be marked up as specified in RFC 3966, if I can manage it.