The bandwidth required for capture and measurement of signals depends greatly on the signals to be measured, the types of measurements to be made, and the accuracy desired of the measurements. A rough rule of thumb most engineers use is to have an oscilloscope with three times the bandwidth of the highest frequency signal they wish to measure, though this becomes impractical for very high frequency signals.
Reference the definition for oscilloscope bandwidth in the FAQ (above). Most oscilloscopes approach the -3 dB bandwidth-rated frequency slowly, beginning with a gentle amplitude rolloff at 50% (or so) of the bandwidth frequency rating. This means that if the oscilloscope amplitude response is -1 dB at 70% of rated bandwidth and -2 dB at 85% of rated bandwidth, then the amplitude of the captured pure sinusoid will be approximately 90% (-1 dB) or 80% (-2 dB) and 70% (-3 dB) compared to when the input sinusoid frequency is approaching the bandwidth rating of the oscilloscope. However, most engineers are not measuring pure sinusoids with their oscilloscope. Note that the highest bandwidth oscilloscopes may have a flatter (less amplitude rolloff) or adjustable amplitude response, for a variety of reasons.
More likely, an engineer is measuring a signal that resembles a square wave. In this case, it is known that a square wave can be represented as a Fourier series expansion comprised of the sum of the fundamental frequency and odd harmonics, with the Nth harmonic contributing a 1/N amplitude at that frequency. What this means is that to accurately represent a square wave, you need enough bandwidth to capture the fundamental frequency and enough of the odd harmonics. How many odd harmonics is “enough” (and how much bandwidth is needed) is determined by the engineer’s tolerance for a rise time measurement on the oscilloscope that is slower than the real signal, and the amount of additive overshoot and ringing present on the measured signal. If only the 3rd harmonic is captured, the rise time will be appreciably slower, and the overshoot and ringing will be noticeable compared to if the 99th harmonic is captured (in which case the captured signal will be indistinguishable from the original input signal).
This gets us back to the original answer that is given most often in response to the question of “how much bandwidth is needed?” – about 3x the bandwidth of the highest frequency signal. But what does “highest frequency” mean? In this context, most engineers are thinking of the rise time measurement capability of the oscilloscope (which is related to bandwidth). If an engineer wants to measure a signal with a rise time of 1 ns, they would not choose an oscilloscope with a 1 ns rise time (such an oscilloscope would typically have a bandwidth of 350 MHz) – they would choose an oscilloscope with bandwidth 3x that (or 1 GHz).
Reference webinar Part 2: How Much Bandwidth Do I Need in My Oscilloscope? in the 2023 Oscilloscope Coffee Break Webinar Series for other details.