Stasi surveillance operations targeted at individuals - suspected dissenters, criminals, traitors, spies, etc. - fell into two general categories. The first was an "Operational Personal Surveillance" (Operative Personenkontrolle, or OPK). OPKs were relatively common; thousands of them were initiated by Stasi officers every year in the 1980s. They required only a modicum of suspicion, did not usually lead to a criminal prosecution, and could have any number of goals. Sometimes, OPKs were initiated to assess whether the target was committing or planning to commit a crime. Sometimes, the goal was actually to recruit the target as a Stasi informant. The level of commitment required of the Stasi officers for an OPK was comparatively low. It might involve nothing more than in-person conversations with a handful of informants. Such seems to have been the case with the OPK launched in 1979-80 against British historian Timothy Garton Ash (codename "Romeo"), who lived in Berlin at the time. The Stasi suspected that he was an MI6 agent; Garton Ash denies this. If you're interested in a close reading of a Stasi OPK file by a brilliant writer and incisive thinker, I recommend Garton Ash's memoir The File.

The second type of Stasi surveillance operation, an "Operational Case" (Operativer Vorgang, or OV) was usually the result of an OPK that "struck gold." If the Stasi discovered that there was something to a suspicion - that a crime or conspiracy was definitely in the offing, for example - an OV could be launched. OVs were larger, more intricate affairs. They might go on for a long time (years), and involve multiple Stasi officers and networks of informants. Surveillance was conducted in person (simply following the target around) as well as over the phone (tapping), optically (cameras), or audially (bugging). Mail was intercepted and read; there was an entire Stasi department ("M") devoted to steaming open envelopes. OVs could produce volumes of paperwork. They also often involved active counter-measures taken against the target, rather than just passive observation. Informants might be infiltrated into the target's social circles. The Stasi practiced what they called Zersetzung ("subversion") - essentially taking steps to hinder the target's activities or simply ruin their life. One of the most vicious and self-aware acts of Zersetzung employed by the Stasi was to spread rumors in a target's community that the target was ... working for the Stasi.

The case portrayed in The Lives of Others is an OV - and a fairly serious one, at that. Playwright Georg Dreyman's apartment is thoroughly bugged. His phone is tapped. The Stasi even wires into the electric lines in his building so that they can ring his doorbell remotely. He is followed by Stasi agents, and a round-the-clock narrative of his activities is written. Then -- SPOILERS! -- his girlfriend, Christa-Maria, is pressured into informing on him. There's even a bit of Zersetzung when Hauptmann Wiesler arranges for Dreyman to learn that Christa-Maria is having an affair with a high-ranking member of the Central Committee.

So, a couple of things should jump out here in relation to your question: First, these OV operations could be incredibly intricate, so it should be no surprise that the bugging/tapping technology used by the Stasi was also incredibly intricate. The Stasi actually preferred, when bugging offices or apartments, to do so from adjoining rooms or apartments, by drilling listening devices into the walls, making them undetectable from inside the target's home. (This also made installation easier, as one did not have to worry about the target discovering the agents in the act.) Listening devices were designed to be as small, unobtrusive, unidentifiable, and inaccessible as possible. Finding one was not, as portrayed in The Lives of Others, as simple as yanking out wiring that bulged suspiciously from underneath the wallpaper. Were the target to discover a bug or tap, there is a good chance they simply wouldn't have known what they were looking at. The average GDR citizen was, after all, not likely to be well-versed in Bulgarian microelectronics.

Second, bugging was not the most common or important way that Stasi officers kept tabs on targets. A Stasi operation would never have relied on bugs to such an extent that the operation would have to be abandoned if one or more of them were discovered.

Regarding your follow-up question, it's important to remember that the Stasi operated outside of the normal law-enforcement apparatus in the GDR. It was often tasked with investigating crimes, but it was simultaneously both more and less than a police force.

Less than a police force: Evidence collected by the Stasi from informants was not directly admissible in criminal courts, and had to be "laundered" through other putative sources. The Stasi was also not empowered to imprison people for long periods of time. The famous Hohenschönhausen prison portrayed in The Lives of Others was in fact a "remand prison" intended for the short- to medium-term confinement of people under investigation or interrogation by the Stasi.

But more than a police force, too: The Stasi didn't need to abide by rules regarding the burden of proof. More or less, if the Stasi wanted to pick you up and talk to you, it could do so. If the Stasi needed that pick-up to be camouflaged, they had special vehicles for the purpose. You might be visited at home - or pulled off the street - to come and "clarify an issue" with the officers of the Ministry. Arrest, for the Stasi, was not the end of the investigation, but rather a continuation of it, as it was inevitably followed by in-person interrogation - a terrifying form of information-gathering at which the Stasi was especially skilled.