A few short stories by Edmond Hamilton:
The Monster-God of Mamurth (An A. Merritt homage)
The Man Who Evolved (inspired the crappy Outer Limits episode "The Sixth Finger")
Fessenden's Worlds
He That Hath Wings
In the World's Dusk (A Clark Ashton Smith homage)
What's It Like Out There?
All of these had the potential to be very good, the basic ideas are very interesting, but the writing ain't that hot.
Next:
City of the Living Dead by Fletcher Pratt & Laurence Manning
The earliest story that involves a form of virtual reality? It's an ok story, but there are some completely random racist things shoe-horned into it that are just bizarre. Just totally out of the blue and with nothing to do with the story. Very weird.
Next, more crappy Hugo nominated short stories:
Still Life by David S. Garnett
Dinner in Audoghast by Bruce Sterling
The Dragon Masters by Jack Vance
Hong's Bluff by William F. Wu
Buffalo by John Kessel
Press Ann by Terry Bisson
Dog's Life by Martha Soukup
Little Dog Gone by Robert F. Young
The Good Pup by Bridget McKenna
The Wedge by Isaac Asimov
Once a Cop by Rick Raphael
Rat Race by Raymond F. Jones
Also:
The Coon Suit by Terry Bisson