This list, though extensive, is in no way exhaustive. Many literary journals will consider short prose whether they advertise that fact or not. But these links lead to journals that have expressed a specific interest:

*82 Review
100 Word Story 

Arts & Letters
Atlas + Alice
Atticus Review

Baltimore Review
Barren Magazine
Bending Genres
Booth Review
Brevity Blog (Craft Discussions, Essays on the Writing Life)
Burningwood Literary

Catholic Digest “Last Word” essays 
Cincinnati Review (micro contest)
Citron Review
Cleaver
Craft Literary
Cutbank

decomP
Deep Wild Journal
Diagram

Ellipsis Zine
Empty House Press

Fairy Tale Review
Feed
Fugue

Gigantic Sequins
Gordon Square Review
Guernica (fewer than 2,500)
Gulf Coast

Hippocampus
Humana Obscura (nature focused)

Ilanot Review
Indiana Review

JMWW
Juked

Long Leaf Review
Lost Balloon
Lunch Ticket

Monkeybicycle
MoonPark Review
Mslexia (UK based; “for women who write”)

New Delta Review
New South Journal
New Orleans Review
NY Times Modern Love (a bit longer, 1500-1700 words)

Passages North
Pidgeonholes
Pithead Chapel
Porter House Review
Proximity

Quarter After Eight

River Teeth (Beautiful Things)

Sidereal Magazine
Sleet Magazine
Smokelong Quarterly
Sonora Review
Split Lip
Spry Lit
Stonecrop Review
Sundog Lit
Sweet: A Literary Confection

Tahoma Literary Review
The Adroit Journal
The Christian Science Monitor’s “Home Forum” section (‘upbeat’ essays)
The Florida Review
The Forge
The Offing
The Palisades Review
The Pinch Journal
The Sun’s “Readers Write” section
TriQuarterly Online

Unbroken Journal
Under the Gum Tree

Waxwing
Wildness
WOW! Women on Writing

X-R-A-Y

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Further Resources

The Best of Brevity: Twenty Groundbreaking Years of Flash Nonfiction

The Rose Metal Press Field Guide to Writing Flash Nonfiction

Crafting the Personal Essay

“Flash Nonfiction: An Interview with Dinty W. Moore” (River Teeth blog)

Literary Magazines and Journals Database  (from Poets & Writers; filter by “creative nonfiction”