Okay—it's time for a real answer. Get this done right the first time around:
PLAY CAMPAIGN—NOT ARCADE or VERSUS. The latter two require "unlocking" executions via multiplayer kills, but there's no such requirement for campaign. I did the following on "NORMAL" and used the Lancer to down the Locust.
ACT 5-5: Ascension
You start with *LANCER*, *GNASHER*, *GORGON PISTOL* (3)—hold down Y over a downed Locust to execute them in style. If you did it right, you'll get a notification in the bottom-right corner. Anya has a *SAWED-OFF* (look at her with LT and press X with your shotgun to trade).
Pick up the *HAMMERBURST*, *BOOMSHOT*, *TORQUE BOW* (3) in the first room (hold down Y). There's a *MORTAR* in the desk area, but you there will be others if you're cramped for time.
After the elevators, there's a *RETRO LANCER* on the right-hand side of the maintenance tunnels. (There's another one later.)
When fighting down the corridor to save someone special, you'll find the *BOOMSHIELD* ("Shield Bash") and a *MULCHER*. Kicking down the doors along the hallway, there's a *RETRO LANCER* on the left and a *ONESHOT* on the right. (4—hold down Y)
ACT 5-6: Reckoning
Make sure you're on at least Normal, otherwise it'll be hard to down the Locust to use the *HAMMER OF DAWN* on them. You can wait for Locust to execute when the Tempest is flailing around (after the cutscene where the Locust grapple up the walls).
There are alternative levels for the Hammer in the comments below.
ACT 2-3: Forced Entry
There's a *LONGSHOT* at the beginning (and a Oneshot if you missed it before). Go ahead and alert the guards. Where the gates open, you'll find *DIGGER*s, a *MORTAR*, and Retros if you missed them before. (3—hold down Y)
ACT 1-6: Hanging by a Thread
You start with a *BOLTOK PISTOL*. Baird has a *SAWED-OFF* if you trade with him (look at him and press X with your shotgun). When the main doors open and the Boomers come out, one will drop a *CLEAVER*. Carry it into the next section for the execution. (3—hold down Y)
ACT 5-2: Blackout
*SCORCHER* is near the beginning (just after the lift stops).
Wasn't that easy? Now let's clean up. Check your Achievements listing in "Stats & Awards" to see what you're missing:
CURB STOMP: Press Y (instead of holding it) near a downed enemy
FACE PUNCH: Hold Y when playing multiplayer/arcade on COG with a weapon that hasn't unlocked special executions yet
ARM RIP: Same, but with Locust
CAPTIVE: Press X to pick up a downed enemy as a meat shield. Press X again to get rid of him.
LANCER CHAINSAW: Hold down B near an enemy with the Lancer (doesn't need to be downed).
RETRO BAYONET: Hold down B and charge into an enemy with the Retro Lancer (doesn't need to be downed).
Find a better location? Let us know in the comments.
Here's a video by TehVGG, if you want (not necessary):