Cable tapping was definitely a big method employed. Undersea cables for example became an excellent target for the US intelligence community to focus on in an effort to glean intel about Soviet plans, technology and operations.

Operation Ivy Bells, initiaited in the early 1970s, is a very well known example I know a bit about it, so will focus on that specific one here to give you a partial answer.

Ivy Bells was a joint operation between the CIA, NSA, and the US Navy, where they employed a purpose-built Navy sub (the USS Halibut) to specifically target a Soviet undersea comms cable line in the Sea of Okhotsk, near the Kamchatka Peninsula. This particular undersea cable ran between the USSR's Pacific Fleet headquarters in Vladivostok to one of their primary fleet bases at Petropavlovsk.

Operation Ivy Bells utilized the USS Halibut sub to sneak into these Soviet territorial waters and drop down on top of the cable, where, submerged in about 400 or so feet of water, the sub then deployed highly trained divers (breathing a helium/oxygen mixture of gases via rebreather dive gear due to the depths involved) to attach a very secretive sensor device to the undersea cable itself. Because the US could not literally cut into the cable and "tap" it directly without possibly setting off alarm bells, instead this 3-foot long sensor device (later upgraded to a longer - about 20-feet in length - device), attached to the exterior of the cable, was designed to be able to "listen" to the electric signals being sent through the undersea cable by analyzing and recording the induction waves the cable gave off as the electric signals passed through it. These signal intercepts were recorded on a tape recorder attached to the device, and once a month or so, the USS Halibut snuck back into the area and again deployed divers to retrieve the tapes and, on occasion, upgrade the signal-detecting sensor equipment. The US (via AT&T's famous Bell Labs) even developed a nuclear-powered RTG version of the sensor, which had a much longer lifetime before needing to have its power source replaced.

Ivy Bells was apparently very successful, going on for over a decade before it was discovered - discovered it is believed thanks to an American NSA employee-turned-turncoat named Ron Pelton, who sold the information about the cable tap to the Soviets for $35,000. While it lasted and prior to its discovery, Ivy Bell was deemed a very effective intel-gathering operation, especially because much of the signals intelligence (SigInt) that the sensor device recorded was being transmitted over this Soviet undersea cable "in the clear", meaning that it was not encrypted. Due to the success of Ivy Bells, other Soviet undersea cables were also targeted around the world, using much the same technique.

That is but one example of SigInt undersea cable tapping/sensor device listening that was employed by the US against the USSR during the Cold War. Other techniques and targets were of course "attacked" to try to glean SigInt data as well, including Soviet satellite systems. A lot of this SigInt spying tech was extremely expensive and usually risky to deploy/execute, and no expense (in money or human lives) was spared by the USA during the Cold War when it came to trying to get intelligence on the Soviets.