The problem is, where to start? There is so much of it.
Well, with the guidance of the Wikipedia (yes, I know) articles on the short fiction categories, and my own notes, I've compiled a list of the places of publication of the short fiction I actually voted for in the last five un-puppied years.
2010
Short Story: Spar, by Kij Johnson (Clarkesworld)
winner was Bridesicle by Will McIntosh (Asimov's)
Novelette: It Takes Two, by Nicola Griffith (Clarkesworld)
winner was The Island, by Peter Watts (The New Space Opera vol 2)
Novella: The God Engines, by John Scalzi (Subterranean Press)
winner was Palimpsest, by Charles Stross
2011
Short Story: For Want of a Nail, by Mary Robinette Kowal (Asimov's) - I voted for the winner
Novelette: Eight Miles, by Sean McMullen (Analog)
winner was The Emperor of Mars, by Allen Steele (Asimov's)
Novella: The Lifecycle of Software Objects, by Ted Chiang (Subterranean Press) - I voted for the winner
2012
Short Story: Movement, by Nancy Fulda (Asimov's)
winner was The Paper Menagerie, by Ken Liu (F&SF)
Novelette: "Six Months, Three Days", by Charlie Jane Anders (Tor.com) - I voted for the winner
Novella: The Man Who Bridged the Mist, by Kij Johnson (Asimov's) - I voted for the winner
2013
Short Story: Mono no Aware by Ken Liu (Lightspeed Magazine) - I voted for the winner
Novelette: "Fade to White", by Catherynne M. Valente (Clarkesworld)
winner was The Girl-Thing Who Went Out For Sushi, by Pat Cadigan (Edge of Infinity)
Novella: The Emperor’s Soul, by Brandon Sanderson (Tachyon Publications) - I voted for the winner
2014
Short Story: The Water That Falls on You from Nowhere, by John Chu (Tor.com) - I voted for the winner
Novelette: The Truth of Fact, the Truth of Feeling, by Ted Chiang (Subterranean Press)
winner was The Lady Astronaut of Mars, by Mary Robinette Kowal (Tor.com)
Novella: Six-Gun Snow White, by Catherynne M. Valente (Tor.com)
winner was Equoid, by Charles Stross (Tor.com)
This really does help me to drill down. In the last five puppy-free years, I voted for three stories published by each of Asimov's Science Fiction, Clarkesworld, Subterranean Press and Tor.com (and both Asimov's and Tor published two winners which I did not myself vote for). If they've published stuff that I liked in the past, it's likely that they will do so in the future. So I'm inclined to make my contribution to the 2016 nominations by reading the short fiction published in those venues this year, along with anything else that should catch my eye in the meantime (such as Penric's Demon, which at present is the only item on my nomination lists for short fiction). I'll also add Strange Horizons, which I always enjoy. Any other recommendations are also welcome.