Remember, this the chronological order in which the stories take place, not the order in which they were written.
1) Conqueror Worm - November 1797 - published in Evolution Expects
2) Unnatural History - April-June 1997 - first published in 2007
3) Leviathan Rising - July-August 1997 - first published in 2008
4) Fruiting Bodies* - September 1997 - published in El Sombra (2007)
5) Vanishing Point* - October 1997 - published in Leviathan Rising
6) Human Nature - November 1997 - first published in 2008
7) Christmas Past - December 1997 - published in Human Nature
8) Evolution Expects - January-February 1998 - first published in 2009
9) Blood Royal - March-April 1998 - first published in 2010
10) White Rabbit* - May 1998 - first published in 2007
11) Proteus Unbound - May 1998 - published in Dark Side
12) Worthless Remains - June 1998 - published in Clockwork Cairo (2017)
13) Dark Side - June 1998 - first published in 2010
14) Anno Frankenstein - September 1943 - first published in 2011
15) Time's Arrow - May 1998 (again) - first published in 2012
16) Tempus Fugit - June 1998 (again) - published in Abaddon X (2015)
17) Reckless Engineering - July 1998 - published in Steampunk International
Stories marked with an asterisk * are also available as part of The Ulysses Quicksilver Short Story Collection eBook. The first three Ulysses Quicksilver novels, and associated short stories and novellas, have been collected together as The Ulysses Quicksilver Omnibus Vol.1, while the fourth, fifth and sixth (and associated short stories and novellas) are now available as The Ulysses Quicksilver Omnibus Vol.2.
You can even pick up the eBook of Ulysses Quicksilver's first adventure Unnatural History for free here! Just remember, "Evolution Expects!"