The Stakeout

A character, group of characters or organization wants to solve a mystery or put a controversy to rest, and they decide the only way to solve it is to pack up and set up camp at a location where they believe something of interest will occur, such as their opponent's headquarters, in hope of finding some critical clue. Often they will go to rather extreme lengths to find just one critical clue. Almost invariably, there's a risk that someone will find out about the operation and blow the observers' cover, and all their efforts will be ruined... or worse.

Usually, it's The Hero's Nakama who are doing this, but occasionally a Villain will use a stakeout, often to find a weakness in their intended victim's security, or reveal their Secret Identity. And it almost certainly happens in Real Life, though probably not as often as in fiction.

Generally, to be a stakeout, the following conditions must be met:

The characters are watching one or more locations.
The characters involved in the operation must hide the fact that they are doing it from someone (or something) else.
The purpose of a stakeout is to gather intelligence. If the purpose is to surprise someone, that's an ambush. If the purpose is to trick someone into revealing something he would't have otherwise, it's a sting. Of course, there's no reason a stakeout can't become a starting point for a sting or ambush operation.

Stakeouts are a fairly universal trope. Common in Detective Drama, they are also found in many other genres including Adventure, Comedy, and even Fantasy. TV shows almost always include this trope during an episode that parodies a cop or detective show. Since it can be a case of Truth in Television,it's an appropriate plot point for shows that are on the realistic end of the Sliding Scale of Realistic Versus Fantastic - but it's also a strategy that might believably still work even in Cloudcuckooland. It's simply applying the idea of being in the right place at the right time in a systematic way. Any character with the patience and self control, and sometimes the stealth and courage, has at least a small chance of being able to pull off a stakeout caper successfully.

It's also easily Played for Drama or Played for Laughs. Usually the question of importance to the plot is not the morality of subjecting someone to such scrutiny, but whether the team will be able to get anything useful before they're stopped by the opposing side, the press, the neighbors, or their own superiors. At any moment, they might find or miss something important or their cover might be blown. The characters lives, careers, and reputations, or the lives of innocents, might be on the line if they're caught or the stakeout proves useless. Having a group of characters stuck in one spot maintaining The Masquerade that nothing unusual is going on is a great opportunity for dialog, character introduction, or character development. It's also a great opportunity for a moment of subtle humor to relieve the tension.

This trope provides plenty of opportunities for the writers to prolong the drama with dramatic or humorous sidestories. Amung the more commonly seen:

If the stakeout lasts for more than a couple days, the group will often add more personnel or equipment as time goes by. If a well - funded agency, that will often include wiretaps, hidden cameras, telescopes, laser - triggered alarms, etc., all to watch their "opponent".
They start snacking, and run out of food, so one character is sent to get more. The others complain about the food.
They get into a debate over something. It's at that moment one of them notices that for once something is happening and tells the others to be quiet.
They temporarily end up spying on someone involved in something titillating but either completely unrelated or only periperally related to the overall mission. If The Hero is spying on a particularly sympathetic relatively innocent person or Anti-Villain, it's likely they'll end up working together later, or else their "victim" will turn out to be Not So Harmless.
The group decides that just watching isn't enough, resulting in a Rear Window Investigation.
Another common result is for one or more of the heroes to be discovered and captured by the villains they are spying on, precipitating the crisis that was delayed during the stakeout.

Super-Trope of Spies in a Van. Often leads to Rear Window Investigation, We Wait.

Examples of The Stakeout include:

Anime & Manga

In Full Metal Panic!, Sousuke, with the help of Kyoko, calls for a stakeout when Kaname goes on a date with an old classmate. Fumoffu.

Film

Matt Helm movie The Wrecking Crew. An ICE agent is sent to watch Count Contini's mansion to determine whether he was involved in the gold heist.
Sneakers. The protagonists do this at least twice: when trying to find out where Dr. Janek keeps his black box decoder, and when trying to find out where the decoder is being kept in the toy company.
The film titled Stakeout
In That Darn Cat, police end up setting up a secret base in a suburban attic and bugging a housecat in their desperation for a clue.
In Tom Clancy's Clear and Present Danger, the CIA rents a flat in downtown Bogota, Columbia to use an evesdropping device.
All the police do in Oscar until the climax.

Literature

The beginning of Frank Herbert's Hellstrom's Hive has a secret agent staking out the title location while posing as a bird watcher. He's hunted down and killed by hive members.
Harry and Murphy of the Dresden Files do this at one point. Given the work, it almost goes without saying that it's played for laughs. Harry, due to total lack of practicing this skill, is bad at it.

Live Action TV

Burn Notice does this often... makes sense, as it's a Spy Show.
Common in The Wire, where the cops lurk in both unmarked cars and abandoned buildings. This is especially the case in the earlier seasons, before the drug dealers wise up and stop using pay phones; if the police are wiretapping a pay phone, they need someone staking out the phone to see who's actually making the call.
Pushing Daisies has a stakeout interrupted by a potato in the tailpipe.
NCIS has had stakeouts on several occasions.
The Rockford Files featured many stakeouts over it's run, but the most unconventional was when the stakeout was completely unrelated to the case the main character was investigating. He simply happens to run into his Friend on the Force disguised as a food vendor. At the end of the episode, he saves himself from the criminals chasing him by leading them through said stakeout.
Young Blades: King Louis joins the Musketeers on a stakeout in "The Chameleon." When he asks what you're supposed to do during one, Ramon produces a tray of beignets.
Showed up from time to time on Dragnet. It was once played for laughs when Officer Gannon started talking about the physical effects he always got with stakeouts-headaches,upset stomach, ect.

Video Games

One dispatch mission in My Sims Agents involves this. What happens depends partially on who you send.
Persona 4
- WHO SAID STEAK

Web Original

Battlecat and Minx (Battlecat's daughter) had a lot of rooftop stake-outs in the Global Guardians PBEM Universe. They'd inevitably descend into discussions about whatever guy she was seeing that week and how Battlecat doesn't think he's good enough for her.

Western Animation

In Justice League Unlimited, Wonder Woman famously starts hitting on Batman while the two are on a stakeout, and she receives a laundry list of reasons why a romance would never work between them.
- And then is in the process of refuting them when they're interrupted.
Some episodes of Recess have TJ and the gang doing this trope, whether its regarding the teachers or even one of their own (the latter only in extreme cases).