The Melthusian captain Bela Tyken encountered a Tyken's Rift at some point in the 24th century or earlier. The rift trapped his ship but he was able to escape by creating a massive explosion using his cargo of anicium and yurium.
In 2367, USS Brattain became trapped in a Tyken's Rift. Suffering from REM sleep depravation, the crew became homicidal and killed each other. Shortly after, USS Enterprise-D almost suffered the same fate. With the assistance of an alien being trapped there as well, the Enterprise managed to break free. (TNG episode: "Night Terrors")
By 2409, Tyken's Rifts had been weaponized. Starfleet, KDF and Romulan Republican science officers trained with the ability were able to employ a ship's main deflector to generate a Tyken's Rift and aiming it at an enemy vessel. This would create a several hundred meter large, bright red fissure in space, sucking power from the afflicted object and damaging it slightly. Gravimetric scientists had a chance to generate more than one Tyken's Rift at a time. Artificial Tyken's Rifts endured for less than a minute. (STO mission: "Welcome to Earth Spacedock")