Introduction:
In the context of detecting a single frequency value, the most sensitive detector is a tuned detector. This answer can be explained based on the working principle and characteristics of the tuned detector compared to other options such as a vibration galvanometer, headphones, and an oscillator.

Tuned Detector:
A tuned detector, also known as a resonant detector, is a device that is designed to respond selectively to a specific frequency. It operates based on the principle of resonance, where the detector's resonant circuit is tuned to match the frequency of the signal being detected. When the frequency of the input signal matches the resonant frequency of the detector, maximum energy is transferred to the detector circuit. This results in a highly sensitive response to the specific frequency of interest.

Vibration Galvanometer:
A vibration galvanometer is an instrument used to measure small electric currents. It consists of a coil suspended between the poles of a permanent magnet. When a current flows through the coil, it experiences a force due to the magnetic field, causing it to vibrate. While a vibration galvanometer can detect current variations, it may not be as sensitive as a tuned detector when it comes to detecting a single frequency value.

Headphones:
Headphones are transducers that convert electrical signals into sound. They are commonly used for audio output. While headphones can reproduce a wide range of frequencies, their sensitivity to a specific frequency depends on their design and specifications. However, compared to a tuned detector, headphones may not be as sensitive in detecting a single frequency value accurately.

Oscillator:
An oscillator is a circuit that generates an output signal with a specific frequency. It is used for various purposes, including signal generation and frequency measurement. While an oscillator can produce a single frequency, it may not be the most sensitive detector for accurately detecting a single frequency value. An oscillator is typically used as a signal source rather than a detector.

Conclusion:
In summary, a tuned detector is the most sensitive detector for detecting a single frequency value. Its resonant circuit is designed to respond selectively to the specific frequency of interest, resulting in a highly sensitive response. Compared to options such as a vibration galvanometer, headphones, and an oscillator, a tuned detector is specifically optimized for this purpose, making it the most suitable choice for detecting a single frequency value accurately.