Electronics hacker “GreatScott!” made a video using a TV CRT as a “crude” oscilloscope.
In the video he shows that the horizontal scan normally runs at some 15kHz, and its return at the end of the scan is not very clean. He demonstrates visualizing waveforms of a few times that frequency.
One difficulty is that the deflection coils in a TV CRT produce a deflection that’s proportional to their current, but he’s driving them with a voltage signal, and they are primarily inductive loads — one consequence is that they more strongly attenuate higher frequencies, but another is that they produce frequency-dependent phase shifts (of nearly 90°) and thus badly deform the waveforms.
It seems like a straightforward solution to this problem is to drive the coils from a current source rather than a voltage source, so that up to some potentially fairly high voltage the magnetic field will faithfully reproduce the desired signal.
Scott’s TV originally used a -42V to +30V signal for the horizontal scan at a bit over 15 kHz (with a voltage waveform that was a distorted square wave, thus producing a distorted triangle wave of current for the scan), with a 600mΩ resistance and 210μH of inductance. The vertical scan used -10 to +40V at 43 Hz, with 21Ω resistance and 20mH of inductance.
In both cases there were strong harmonics in the signals.
It seems to me that if you want to make this a useful oscilloscope tube, your best bet is to turn the screen sideways, because the “vertical” deflection of your signal basically always needs to have higher frequency components than the “horizontal” scan. If a ±40V signal is adequate to recognizably reproduce a deflection of some 150kHz, then by using a current source with compliance up to 40kV, we should be able to successfully reproduce 150MHz. The coil probably would not survive 40kV, though, as special coil construction is needed to handle voltages over a few hundred volts, so we could probably only reach a few MHz in practice without rewinding the coil.
At the 15kHz fundamental, the 210μH inductance represents about 20 ohms (making its 0.6Ω real resistance insignificant at that frequency or above), so getting a full deflection requires on the order of an amp or two through the coil.
Purpose-designed oscilloscope tubes differ in a few ways: they use electrostatic deflection, which I think is not a thing you can retrofit to the tube afterwards; are longer so that smaller angles are feasible; and have an axial voltage gradient to progressively accelerate the beam so that the electrons are moving more slowly as they go through the deflection plates than when they hit the screen. (In part this is to compensate for some disadvantages of electrostatic deflection: it deflects fast electrons less than slow ones, so the speed must be relatively low and consistent across all the electrons in order to get a large and consistent deflection. Magnetic deflection does not have these problems; the radius of the electron track depends only on the magnetic flux density, not on its speed.)
A computer monitor might be more promising. This TV had only about 350 lines per scan and a very low refresh frequency, so it could get by with a low horizontal scan frequency. By contrast, the LG 710e monitor someone threw out the other day is reported to support 1024×768 at 85Hz. This means that its normal horizontal scan must run at at least 65kHz. I don’t know if it uses a higher voltage than ±40V or not; it would seem like using a coil with half as many turns would give you a quarter the inductance but also half the field, which would be a win.
(See also files VCR oscilloscope, Laser printer oscilloscope, and CCD oscilloscope.)