SpongeBob is walking home with a very heavy bag of snail food for Gary, and Patrick tells him that he has something important to tell him, but SpongeBob ignores him and passes Patrick's House before he can say it. At his front door, SpongeBob finds a package. He leaves the snail food on the porch and opens the package to find a Mermaid Man and Barnacle Boy paddleball set. SpongeBob begins to play with it, and finds the "Dirty Bubble Challenge" inside the box, which presents him with the absurdly impossible task of hitting the paddleball 29,998,559,671,349 times in a row. He accepts the challenge, and repeatedly attempts it, never being able to successfully hit the ball even once. Meanwhile, Gary goes to his food bowl to eat, but finds it empty. He confronts SpongeBob for his food, but SpongeBob is too preoccupied with the paddleball and tosses Gary aside. Feeling disowned, Gary abandons SpongeBob, who does not even notice as he leaves. Gary wanders across the sea aimlessly, eventually departing on a bus to parts unknown.
Sometime later, Patrick goes to SpongeBob's House to tell him what he was going to say, remarking that he hasn't seen him in a couple of days. He finds that the house has deteriorated into an unkempt mess, and finds a psychologically broken SpongeBob lying in fetal position in his private quarters. SpongeBob tells him that he became this way from taking the Dirty Bubble Challenge, and Patrick wails at the mention of the challenge, saying that he beat the challenge years ago and won, but then he lost the trophy they sent. SpongeBob, who has now gotten over the challenge, invites Patrick into the kitchen for cookies and milk. He gives Patrick a "fresh" glass of milk, and trips over Gary's food bowl. He remembers that he must feed Gary, and retrieves the bag of food from his porch, remarking that it has gathered a lot of dust for "a couple hours." When he calls Gary for dinner, and when he does not come, SpongeBob asks Patrick how long he was taking the Dirty Bubble Challenge. Patrick, judging by the condition of the milk, which has spoiled to the point of being completely solid, tells SpongeBob that he was gone for about seven to ten days. SpongeBob begins desperately looking for Gary, until he checks in the mailbox, where he finds a note from Gary telling him that he has run away in search of a new owner that will remember to feed him. SpongeBob is devastated by this.
Meanwhile, Gary comes to a seedy and uninviting neighborhood, and after a close encounter with a gang of alley snails (who just wanted to give him nachos), Gary is found by an old lady who recognizes him as "Ms. Tuftsy." She takes Gary into her house, where she pampers and overfeeds him. Meanwhile, SpongeBob desperately searches all over Bikini Bottom for Gary, but to no avail. The next morning at work, SpongeBob is extremely depressed and unproductive, and Mr. Krabs tells him that "any problem you have can be solved with a little hard work." SpongeBob interprets this as meaning that he must work harder to find Gary, and with newfound confidence, leaves the Krusty Krab to continue his search. He puts up fliers around town that directly apologize to Gary and ask him to come home. They put one on the door of a crafts store, and when Patrick sees that the store is having a sale on scented pine cones, he rushes in to get some. Inside, he sees Gary with Grandma, and asks her where the scented pine cones are. She tells Patrick that she bought all of them, and Patrick leaves, oblivious to the fact that he has found Gary and not mentioning it to SpongeBob.
This is followed by a song set to a montage of a heartbroken SpongeBob (along with Patrick) unsuccessfully searching for Gary and publishing media asking him to come home (including a billboard, a television announcement, and skywriting). After returning from a day out with his new owner, Gary indicates that has to go to the bathroom, and Grandma gives him a stack of fliers given to her by "a little chubby boy." Gary sees that the stack of flyers is from SpongeBob, and realizes that he does indeed care about him. Gary feels sorry for SpongeBob, so he tries to make him feel better by returning. While Grandma is in the kitchen getting the meatloaf she had been cooking for him, Gary tries to find an exit, and stumbles across a closet full of dead snail shells. Gary realizes that Grandma is overfeeding him and presumably did this with many other snails before. Grandma chases Gary, and now outside, He hides in a back alley, where he encounters one of the alley snails he met earlier. He pushes him into the open, where Grandma believes him to be her snail and takes him in place of Gary. Meanwhile, SpongeBob has given up on looking for Gary, and wishes to see him one last time, when Gary suddenly comes to him. SpongeBob promises to never neglect him again, and takes him inside to feed Gary, saying that he must be "starving," unaware of what Gary has been through, which Gary responds with a denial meow stating that he's full.
In a press release on November 1, 2005, Nickelodeon officially announced the episode and promoted it as the "Where's Gary" special. According to the network, "It's a sad day in Bikini Bottom when SpongeBob learns that his beloved pet snail Gary is missing and that he may be at fault in the mollusk's disappearance." Marjorie Cohn, Executive Vice President of Development and Original Programming for Nickelodeon, said "Kids love their pets and there's certainly some relatability here in the relationship between SpongeBob and Gary -- skewed though it may be[...] It's fun to see Gary express a whole range of emotions using only his signature 'meow.'"
According to the storyboard, SpongeBob was originally going to visit Squidward's house and ask Squidward if he could give him a hand with the bag of snail food. This was either cut to save time or it was replaced with Patrick.
"Have You Seen This Snail?" was viewed by eight million people. It was the highest-rated program on all TV with children aged two to eleven for the year of 2005 behind the Super Bowl and the Super Bowl kick-off, and the highest-rated program on all of cable TV with children aged two to eleven and children aged six to eleven in 2005.
Critics produced mixed reviews for Have You Seen This Snail?. Tom Shales of The Washington Post called the episode deeply hilarious and described the plot as a "bittersweet riot." He drew attention to how the story is similar to the plot of "Dumped," an earlier episode in which Gary deserted SpongeBob.
The television critic for the St Petersburg Times was not so positive about the episode however, criticizing how SpongeBob's disappointed and sad character for "Have You Seen This Snail?" is quite different to the character's normal bright personality.
Nickelodeon supported the episode with a month-long off-air marketing campaign including print, outdoor and consumer product partner support, along with the release of a preview clip and bonus coverage of the episode on Nickelodeon's broadband online platform TurboNick, and the Trail of the Snail online game.
In a sweepstakes event with the promotional name of the special, every night for five days leading to the episode's premiere, Patchy the Pirate, and other celebrity guests searched in various locations for Gary. By visiting Nick.com, the viewers could receive a "Trail of The Snail" tracking card to enter Patchy's search locations in order to win the sweepstakes. The prize was a trip to the Nickelodeon Animation Studios to meet the production team of the show and to sit in on a voice recording session with the series cast.
Oddly one of the buses was seen being driven by a police fish rather than a typical Bikini Bottomite.
The original plot for this episode was Granny is fattening Gary up to eat him, but that was changed due to it being deemed too scary for a children's series. Also, she had an axe that was replaced with a tray of cookies.
The original plot is highly similar to the fictional story Hansel And Gretel.
It is revealed that Gary has a clock inside his shell that tells what time he eats.
The title card is very similar to "Wet Painters" due to the painted letters.
When Patrick sees something that is really important, he gets worried by a useless reason.
Gramma calling Gary Mrs. Tuffsy
SpongeBob says "darn it" when trying the challenge.
The scene where Squidward screams when SpongeBob and Patrick enter his house has become a meme on YouTube. The videos are usually named "[Insert name here] drops by Squidward's house." It consists of a collection of videos depicting the scene being edited so that a completely random character is pasted into the picture, covering up view of SpongeBob and Patrick. He screams normally, but it is looped backwards to help extend the video.
Around the time this episode aired, Cartoon Network had a snail resembling Gary at the bottom of the screen during its programs. It should be noted that Cartoon Network and Nickelodeon were bitter enemies during the 90's and early 2000's (due to Cartoon Network's bigger library), so they started insulting each other in their bumpers. 
Gramma seems to be a parody of the witch from Hansel and Gretel, who overfed the children with the intention of eating them. She is also similar to Bluebeard, who kept the corpses of his murdered wives in a closet, which gives away his secret to his new wife.
When Gary and Gramma are driving in the car, Gramma says "Set phasers on fun!," a reference to the Star Trek line "Set phasers on stun."
The song, "Gary Come Home" is remarkably similar to Heart of Gold by Neil Young.
The title is a reference to the quote, "Have you seen this pet?"
The music that played when SpongeBob said "I think it has something to do with this bowl" was a reference to Jeopardy.
When Gramma takes Gary to dance, the blanket is still there, but after that, the blanket is gone.
When Patrick flips SpongeBob, the paddleball is missing, later it reappears.
When SpongeBob trips on Gary's bowl the milk splashes out from the jug, but when Patrick shakes the milk out of the glass it is in a solid form.
When Patrick causes the plane to crash, SpongeBob says "Pat, no!," but captions say "Aah, no!."
When SpongeBob is reading the note left on his door, he walks inside and leaves the door open. But when the mailfish rings the doorbell, SpongeBob opens the door, though it was already open.
When Gary is getting out of the food dish, he doesn't have the running away sack. In the scene right after, he has it on his back.
When Squidward says "What are those neanderthals up to?," his teeth look different and you can see each individual tooth, similar to Patrick's teeth.
When SpongeBob opens the mailbox, it says "Squarepat" when should be "SquarePants."
When Gramma puts Gary down, Gary is missing a dot on his shell.
When Gary gets back to the pineapple, he groans after SpongeBob suggests Gary being hungry. However, he should be hungry since it would have taken him a week to ten days to get back to Bikini Bottom.
Patrick suggests that they were at Martha's Craft Zone that day. However, Gary would have taken a week to 10 days to get back to Bikini Bottom.
SpongeBob is seen driving a plane during the montage of "Gary Come Home" even though he doesn't have his license.