Very, very few historical lutes are in playing condition; the back and neck of a very old lute might well be in stable condition but lute soundboards are thin and usually could not withstand the tension of the strings.
Most historic lutes are in museums, a few in the hands of private collectors.
A new interactive online database of historic lutes that have survived into modern times. Based on data collated over decades by David Van Edwards, Klaus Martius, Friedemann Hellwig, Ernst Pohlmann and others, with many photos and details about these remarkable historical survivors. See help page and summary www.lutesociety.org/pages/extant-lutes-database or go to the database www.lutesociety.org/extant-lutes-db/index.html
Musical Instrument Museums Online, large open database with pictures and information on instruments in museums around the world, including hundreds of lutes: www.mimo-international.com/MIMO/accueil-ermes.aspx
The International Directory of Musical Instrument Collections, CIMCIM, has a searchable database of historical instruments: www.cimcim2009.org/cimcim/ including links to many websites, plus details of CIMCIM publications and an index of construction plans available for instruments in public collections.
Lynda’s Sayce’s theorbo site has links to the websites of museums which hold extended necked lutes: www.theorbo.com/theorboinformation/theorboinformation/museums.html
See also lists on lute history page of this portal.
Most of these have both a website, possibly with pictures and details of historical lutes, and of course publish printed catalogues, which can be purchased online.
Vienna, Kunsthistorisches Museum www.khm.at/besuchen/sammlungen/sammlung-alter-musikinstrumente/
Brussels, Muziekinstrumentenmuseum, www.mim.be/
Copenhagen, Musikhistorisk Museum natmus.dk/en/the-danish-music-museum/
Paris Cite de la Musique: www.citedelamusique.fr/francais/
Online catalogue: www.citedelamusique.fr/francais/musee/visiter/catalogue.aspx
Printed Catalogue of lutes on sale at: www.citedelamusique.fr/francais/musee/publications/publications_scientifiques.aspx
The Fuessen Heritage Museum:
Nuremberg, Germanisches National Museum.
Bologna, Musical Instrument Museum: www.museibologna.it/musica/documenti/76438
Milan, Castello Sforzesco: www.comune.milano.it/dseserver/webcity/documenti.nsf/webal/5A4CA30F4FDC7CA8C125726C005484BE
Rome, Museo degli Strumenti Musicali, www.museostrumentimusicali.it/
Barcelona, Museu de la Música w110.bcn.cat/portal/site/MuseuDeLaMusica
Siguenza, Centro de la Vihuela de Mano y la Guitarra Española “José Luis Romanillos”: romanillos.org/
Stockholm, Musikmuseet, musikverket.se/scenkonstmuseet/
Stockholm, Nydahl collection, www.nydahlcoll.se/
Edinburgh University Collection of Historic Musical Instruments.
London, Fenton House, www.nationaltrust.org.uk/fenton-house/
London, Horniman Museum, www.horniman.ac.uk/visit/displays/music-gallery#image-3
The Horniman Museum is also the postal address for the Musical Instruments Resource Network, offering guidance and support for those maintaining historical music collections, www.mirn/org.uk
London, Royal College of Music, www.rcm.ac.uk/events/museum/
London, Victoria and Albert Museum (instruments currently in store) www.vam.ac.uk/contentapi/search/?q=musical%20instrument#?q=musical%20instrument
Oxford, Ashmolean Museum: www.ashmoleaninstruments.com/gallery.php
National Trust Collections (numerous locations) www.nationaltrustcollections.org.uk/results?SearchTerms=Lute
Historical Lutes in North American museums are catalogued on the links pages of the Lute Society of America website: lutesocietyofamerica.org/resources/external-resources/historical-lutes-and-related-instruments-in-north-american-museums/
Boston, Museum of Fine Arts www.mfa.org/collections/musical-instruments
New Haven, Connecticut, Yale University Collection of Musical Instruments www.yale.edu/musicalinstruments/
New York, Metropolitan Museum of Art, www.metmuseum.org/
Vermillion, SD, National Music Museum, orgs.usd.edu/nmm/Pluckedstrings.html