Sunglasses on, bottoms up. Here's a quick list.

Downtown

Grind City Brewing taproom - *dog friendly
Cordelia's Market Harbor Town *dog friendly actually has two patios
Loflin Yard *dog friendly (the whole place is basically a patio)
Comeback Coffee
The Rendezvous  the new outdoor dining option in their alley way
Cupcake Cutie (just a few tables) 
Curfew at Canopy by Hilton Hotel *dog friendly
Central BBQ Downtown *midtown and downtown locations are dog friendly
Automatic Slim's (just a few tables) 
Alcenia's
Cocozza American Italian (covered and uncovered areas) *dog friendly
Terrace at the River Inn (the covered rooftop bar/restaurant at The River Inn at Harbor Town)
Blue Monkey Downtown (smoking allowed)
Carolina Watershed *dog friendly
Old Dominick Distillery (call ahead for rooftop availability)
The Greyhound at Hilton Garden Inn *dog friendly
Slider Inn - Downtown *dog friendly
Wiseacre Taproom (downtown location) *dog friendly in the outside area only, you'll need to bring someone to hold your dog when you go to the bar inside, health dept. rules. 
Silly Goose *dog friendly
Barware *dog friendly
Talk Shop at Caption by Hyatt Beer Garden *dog friendly
CIMAS at the Hyatt Centric *call for dog friendly info (call ahead to make sure it's open)
Crazy Gander Coffee Co. *dog friendly (coffee shop, a couple of tables)
Raw Girls (two small tables)
By The Brewery
South Point Grocery *dog friendly (covered)
Talk Shop at Caption by Hyatt (covered)

Main Street

Sage Memphis
Cafe Keough 
Maciel's *dog friendly (just a few tables) 
Aldo's Pizza Pies - Downtown *dog friendly
Local Gastropub *dog friendly
Blind Bear Speakeasy (might be some smoking)
Blue Fin
Majestic Grille *dog friendly, covered
The Lobbyist *dog friendly, covered areas
South of Beale *dog friendly
Bedrock Eats & Sweet *dog friendly
Good Fortune Co.  
Green Beetle *dog friendly
Bluff City Coffee
Max's Sports Bar *dogs allowed on front patio only
Ghost River Brewing Taproom *dog friendly
Hu. Hotel's Hu.Roof
Hustle & Dough / Bar Hustle at the ARRIVE Hotel *dog friendly inside and outside
Beck & Call Lounge (rooftop bar at the Hyatt Centric Hotel) reservations recommended
Central Station / Eight & Sand Lounge *dog friendly
IBIS 
WKND Hang Suite (covered)

Beale Street

Silky O'Sullivan's heated patio
Jerry Lee Lewis' Cafe
Alfred's on Beale 
Rum Boogie Cafe
B.B. King's
Tin Roof
Ghost River/New Wing Order 

Cafe Ole

Crosstown / The Edge / Medical District

High Cotton Brewing taproom *dog friendly
Edge Alley
Crosstown Brewing Co. *dog friendly
Crosstown Concourse including Global Cafe, MEMPops, Wolf River Brisket, Abner's (multiple patios, *dog friendly)
Sunrise Memphis
Inkwell 

Midtown + whitehaven

Cafe 1912 *dog friendly, reservations encouraged and recommended
Bari Ristorante *reservations recommended
ECCO on Overton Park *dog friendly, heated patio, reservations required
Otherlands Coffee Bar  *dog friendly
Slider Inn - Midtown *dog friendly, heated patio
Cafe Eclectic Midtown
Huey's Midtown
Railgarten *dog friendly
I Love Juice Bar
Blue Monkey Midtown (might be some smoking)
Hattie B's
Tamboli's Pasta and Pizza
Frida's Restaurante Mexicano 
Hernando's Hide-A-Way

Cooper Young

(most everywhere has at least a few tables outside)
Memphis Made Brewing taproom *dog friendly
Beauty Shop / Bar DKDC *dog friendly (call ahead for details)
Tsunami 
Hammer & Ale *dog friendly
Mulan *dog friendly
Alchemy *dog friendly
Young Ave. Deli *dog friendly (might be some smoking)
Cafe Ole *dog friendly (might be some smoking)
Celtic Crossing *dog friendly (might be some smoking) heated patio
Parish Grocery
Bain BBQ & Bakery
Cooper House Project *dog friendly
Maciel's Midtown Location

Loflin Yard

Overton Square

Babalu
Second Line *dog friendly, please inform the hostess you have your furry friend with you 
Memphis Pizza Cafe Midtown (2087 Madison Ave.) *dog friendly
Bayou Bar and Grill
Boscos reservations recommended
Local on the Square 
Side Street Grill (might be some smoking)
Lafayette's Music Room, heated patio
Robata *dog friendly
Saltwater Crab *dog friendly
Tiger & Peacock (rooftop lounge at The Memphian Hotel) reservations required, dress code, minimum spend
Porch & Parlor (covered)
TJ Mulligan's

Broad Avenue and Binghampton

Maximo's on Broad
Inspire Community Cafe call ahead for patio availability
City & State *dog friendly
Salt | Soy *dog friendly
Wiseacre Taproom *very dog friendly, like, so many dogs
The Cove *dog friendly
The Liquor Store *dog friendly
Bounty on Broad *call for dog-friendly info
Hampline Brewing *dog-friendly

Broad Ave. Wiseacre Taproom

East Memphis

Sidecar Cafe
River Oaks, heated patio (call ahead for reservations)
Big Bill's BBQ, Kirby Rd. locations
Soul Fish at Wolfchase
Libro at Novel in Laurelwood, heated patio
Mosa Asian Bistro
Lost Pizza Co. *dog friendly
Central BBQ  East Memphis Poplar location, *dog friendly, heated patio
Cheffie's (High Point and Shelby Farms locations) *dog friendly
High Point Pizza *dog friendly
French Truck Highland Row location  *dog friendly
RP Tracks (likely be some smoking)
Swanky's Taco Shop (4770 Poplar Avenue)
Memphis Sports Pub *dog friendly
Memphis Pizza Cafe Park Ave. (5061 Park Ave.) *dog friendly
Elwood's Shack *dog friendly
Ciao Bella 
Las Delicias on Quince
Newby's
The Bluff
La Michoacana
ACRE, heated patio, call ahead for reservations
Ronnie Grisanti's
Gus' World Famous Fried Chicken East Memphis
Lucchesi's Beer Garden *dog friendly
Owen Brennan's, heated patio
Patrick's, heated patio
Los Jarochos, Summer Ave. location has patio
Mortimer's *dog friendly
City Silo (Sanderlin location)
Mellow Mushroom Eastgate**, heated patio
Hen House, *dog-friendly
Magnolia & May
Pimento's on Poplar
Venice Kitchen
Tenero Cafe & Butcher 
Rotolo's Pizzeria 

Brookhaven Circle  

Brookhaven Pub (695 West Brookhaven Circle) *dog friendly, heated patio
Andrew Michael Italian Kitchen, heated patio (call ahead for reservations)
Hog & Hominy, heated patio, reservations required
Bogie's Deli Brookhaven
Fox Ridge Pizza
Bangkok Alley 

Hammer & Ale

Germantown + Cordova + Bartlett

- The Half Shell Southwind (7825 Winchester Rd.)
- Forest Hill Grill in Germantown
- Biscuits & Jams 901 in Bartlett, reservations required
- Wolf River Brisket in Germantown
- Huey's - pretty much all the Huey's
- East End Grill - both Bartlett and Southwind locations
- Casablanca (7609 Poplar Pike)
- Mesquite Chop House in Southaven
- Mellow Mushroom in Germantown (9155 Poplar)** *dog friendly
- Kooky Canuck in Cordova (1250 N. Germantown Parkway)
- Flying Saucer in Cordova** dog friendly
- My Favorite Place in Cordova
- Three Guys Pizza
- Meddlesome Brewing *dog friendly patio
- El Mezcal in Bartlett
- El Porton in Bartlett
- Bahama Breeze** at Wolfchase
- Los Cabos in Bartlett
- El Idolo in Bartlett
- El Patron in Bartlett
- Side Porch Steakhouse in Bartlett, covered
- El Mero Taco in Cordova
- City Silo in Germantown
- Rollin Cafe in Cordova
- Jerry's Sno Cones in Cordova
- Sidecar Cafe in Bartlett area 

Collierville

- Tacos 4 Life** 
- Pyro's 
- Cheffie's
- Tony's Trophy Room 
- Cajun Catfish Company 
- Square Beans
- Bangkok Alley 
- Highlander Pub 
- Dyer's 
- Frida's 
- Cafe Piazza 
- David Grisanti's
- Booya's
- Ciao Baby

** I don't normally include non-local chains in my lists of Memphis patios, but so many people suggested these patios that I added them.

How many patios are too many patios for a list of MidSouth patios?