New York, New York City: The American Museum of Natural History.  A great diversity of dinosaurs (representing stegosaurs, ankylosaurs, ceratopsians, pachychephalosaurs, many clades of ornithopod, prosauropods, sauropods (the only real dinosaurian weak spot for the AMNH compared to the Smithsonian and Carnegie, which have both diplodocoids and macronarian skeletons on display), and LOTS of theropods), classic taxa and specimens (including several ornithischians with skin impressions), extremely well lit.  Also present are the truly unsurpassed Hall of Paraphyly... er, Hall of Vertebrate Origins and the mammal halls.  The single best collection  and the single best set of vertebrate paleontology exhibits on the planet. www.amnh.org

Illinois, Chicago: Field Museum of Natural History.     Highlights: Visit the newly unveiled Tyrannosaurus Rex, Sue, at the Field Museum on Natural History in Chicgo and on the web. http://www.fmnh.org/sue  See Brachiosaurus altithorax, Apatosaurus, Daspletosaurus mislabelled as Albertosaurus (suffering from a rather weird homeotic mutation of the pedal digits... :-), Lambeosaurus, Herrerasaurus, and more.  Excellent stuff, and again in a city with a lot else to do.
http://www.fmnh.org/

Ohio, Cleveland: Cleveland Museum of Natural History. Highlights: Vertebrate Paleontology http://www.cmnh.org/research/vertpaleo
http://www.cmnh.org

Pennsylvania, Philadelphia:  Academy of Natural Sciences.  The Academy, the oldest natural history museum in the United States, boasts a dinosaur hall, featuring Giganotasaurus, T-rex, Corythasaurus, and many other casts and fossil dinos, a working paleo lab, a video "Time Machine" (which allows visitors to view themselves on a screen with dinos in a "living" Cretaceous setting.), a Mesazoic "Western Interior Seaway" featuring a Tylosaurus, Elasmasaurus and various sealife, and an extensive collection behind the scenes.  The head of the Vert Paleo area is paleontologist Dr.Ted Daeschler. (A Devonian fish specialist)  www.acnatsci.org

Pennsylvania, Pittsburgh: Carnegie Museum of Natural History.  Highlights: Detailed exhibits see http://www.clpgh.org/cmnh/doe/ for educational resources and the Discovery Room Online with hands-on activities for kids.
http://www.clpgh.org/cmnh

South Dakota, Hill City: Black Hills Museum of Natural History and the Black Hills Institute of Geological Research. Many Cretaceous dinos and more.  Excavated 8 Trexes in the recent past.
http://www.ahaha.com/bhmnh.html,  http://www.bhigr.com/

Utah & Colorado: , Dinosaur National Monument: A museum and a dig all in one!  http://www.americansouthwest.net/utah/dinosaur/national_monument.html   http://www.cr.nps.gov/museum/exhibits/dino/index.html   http://www.utah.com/nationalsites/dinosaur.htm

Washington, Seattle:  The Burke Museum at the University of Washington.  The Burke Museum on the campus of the University of Washington has a number of dinosaur skeletons and casts, as well as a mosasaur, a plesiosaur skeleton cast, troodont nest, plus fossil mammals of various kinds (early baleen whale, mammoth, Smilodon, cast of bloated rhino buried  in lava flow ).  Not a huge selection but definitely worth visiting! The Pacific Science Center also has some dinosaur stuff.  http://depts.washington.edu/vertp/

Wisconson, Milwaukee: Milwaukee Public Museum . Highlights: Excellent on-line tour. "Third Planet" exhibit includes general material on North America plate tectonics, geologic time periods, dinosaurs and mammals. Check out the dinosaur and geology menus. Articles on dinosaur extinction, recreating fossils for display, Pteranodon http://mpm1.mpm.edu/exhibit/third/tp0.html
http://mpm1.mpm.edu/

Wyoming, Casper: Tate Museum (Casper College). Highlights: Newsletter, kids' art and questions.
http://www.cc.whecn.edu/tate/webpage.htm

Wyoming, Laramie: University of Wyoming Geological Museum. Highlights: Features on Allosaurus, Apatosaurus, Pteranodon, Triceratops, Tyrannosaurus rex, and Ctenacodon brentbaatar (new mammal species). The University of Wyoming Geological Museum in Laramie functions to support both public education and scientific research. Wyoming is rich in geologic treasures and the Museum presents to the visitor some glimpse of this geologic diversity. Housing more than 50,000 cataloged fossil, rock, and mineral specimens, it is an important source of information for researchers throughout the world.
http://www.uwyo.edu/geomuseum

Wyoming,Thermopolis: The Wyoming Dinosaur Center and Dig Sites.  The Wyoming Dinosaur Center is a world class museum. It has 12,000 square feet of exhibition area. Fossils and life-forms from earliest geologic time periods are displayed in a time-perspective. There are over 200 displays throughout the museum. The central hall houses 20 full-size mounted skeletons, including 10 dinosaurs.Dig Site Tours - Take a tour bus to visit a world-class discovery just outside of Thermopolis, Wyoming. Dig For A Day - Join the Wyoming Dinosaur Center staff on the hill. Be a part of the excavation process. Kids' Dig Programs - A fun educational experience for the kids!wdinoc@wyodino.org

International Museums

Canada, Drumheller, Alberta: Royal Tyrell Museum. The Tyrrell is a bit hard to get to (fly into Calgary, drive for an hour and a half to the little town of Drumheller). However, given that, these are phenomenal exhibits of Late Cretaceous Canadian fossils, and are smack dab in the rocks whence the fossils came! (I heard a rumor that at the 1988 SVP meeting, Jack Horner was walking around the parking lot and came across a hadrosaur jaw...).  There are exhibits of other dinosaurs here, too.  For researchers on Late Cretaceous dinosaurs of North America, the Tyrrell is Mecca, but that is as much for behind-the-scenes stuff as anything else.   See also the Burgess Shale Virtual Exhibit http://www.tyrellmuseum.com/bshale/index.htm  http://www.tyrellmuseum.com

England, London: Natural History Museum.
Highlights: Works closely with the University of California-Berkley to maintain the Paleonet which serves the professional paleontology community.
http://www.nhm.ac.uk

Germany, Munich: Munich Paleontological Museum.
Highlights: Features a virtual tour of fossils including exhibits ranging from proto-mammals to pteranodons. Check out the "fossil of the month" exhibit while you're there.
http://www.palaeo.de/PMM_Home.htm