12-string acoustic guitar. Models from 1999 onward have a newer style bolt-on neck, and current models all feature a cutaway.
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One of these went up under the Pumpkins' official eBay account on September 7, 2011, ending one week later at $2,750, reserve not met. The guitar was re-listed on October 7, 2011, ending on October 17. The auction listing included the following note:

1997 Taylor 855 12-string Acoustic Electric Guitar. Serial #970418136. Used by Billy Corgan on the Smashing Pumpkins 1998 Adore album and tour, 2000 Machina album and tour, 2003 Zwan Mary Star of the Sea album and tour, and the 2005 Future Embrace album sessions. Billy Corgan has personally signed the guitar. Guitar has gold hardware and a Fishman Prefix Onboard Blender pickup installed. There are minor nicks and scratches due to the touring history of the guitar. A Taylor hardshell case is included. A certificate of authenticity is also included.

Unless the band was secretly doing a lot of KC & The Sunshine covers, the sticker labeled "That's The Way" on the back of the headstock would indicate that the guitar was also used on Zeitgeist.