CCD cameras use what's called "global shutter" which means all the pixels are exposed at once. CCD cameras use "rolling shutter" in all but very expensive professional equipment.

Somewhat older point and shoot digital cameras are CCD. Camera phones and digital cameras from the last few years are CMOS.

It would work better if you can get your hands on CCD based camera when you're dealing with line frequency discharge lamps to avoid the interlace from showing up like this. These lines are characteristic CMOS artifacts as line cycles are captured in interlacing.

The reason is, the CMOS sensor can not store the integrated light information till it is read out (it would cause too high noise), so the shutter has to just precede the area read out. And as the transfer rate is limited, it takes few 10's of ms to read out the complete array, mainly if it is of some very high resolution.

There is quite nice trick to overcome that: Do not read the pixels in a direct sequence as they are physically placed on the die, but e.g. every 4'th pixel horizontally and 4'th row vertically. After going through the complete array, shift that pattern and in that way read the sensor in 16 steps, so such flicker is then averaged out among multiple pixels (usually by the software).
The problem is, when the sensor electronic has to support this mode, the sequencer logic becomes more complex, so requires either denser process (and that means worse analog parameters, so worse noise), or larger gaps between the active pixels, so either the resolution would be worse, or the physical size of the pixel sensor would be smaller, resulting into larger noise.
Or to combat this, you end up with either stricter selection criteria at the end of line test, what means higher scrap rate, so higher cost of the product.

Just use a camera with a proper mechanical shutter

That would make the same effects as with the CMOS, only the sweep is horizontal and not vertical (so the lines from the stroboscopic effect would be vertical and not horizontal), maybe the sweeping speed a bit different.
In fact there is no difference in acquiring the image at all, whether the shutter is optical (blocking the light path) or electronic (controlling the light integration time by an electronic switch on the sensor chip). The only advantage of the mechanical shutter is, the sensor is not heated up (e.g. by direct sun, before you snap the picture), so then for the main picture acquisition it is colder, so could perform better by itself.
But the heating is these days well compensated in the software post processing (before stored on the memory card - it keeps track of such strong light, so sensor heat source; along with the other sensor corrections). So a mechanical device is then only just another complication, so source of problems...
With some models it is still offered for art-photographers, not because it would bring any picture technical quality, but just because the shutter related artifacts are frequently utilized in artistic compositions.

Otherwise the strobing effect was not that much present on film, because the film needs (and tolerates) way higher light exposure, so the opening time was usually rather long (compare to the electronic sensors), so the opening time was longer than the mains period.

This doesn't appear on any of my images of lamps running. Either compact or SLR.

When you manage to get the exposure time in the multiply of 10ms, the effect won't be there, regardless of the sensor system.

The same as FSB - I tried various exposure times (e.g. 1/80 sec.) and the magnetic ballasted fluorescent showed no such strips (several test images done).
Now I tested it with my vintage digital camera from 1997 - there's just a full-auto mode available but several test images haven't shown those coloured strips.

The 10ms is 1/100...
With the multiples it is eliminated completely, but even if it is just longer than that, the effect is greatly supressed.
The effect become quite strong with recent cameras, because the new sensors suffice with exposure times in the 1ms (so 1/1000) range, but the readout time is 100's of ms

Just tried it at 1/1000, and then in 1/1000 increments up to 1/8000. Nothing.

It means either the image is captured at once (so most likely CCD), or the sensor uses interleaved readout sequence (so the bars are there, but with neighboring pixels in opposite phase, so the post processing in the camera SW could even that out before assembling the final picture)
Then you get either darker or lighter image (depend in what phase you happen to be in the flicker pattern), but without any lines or so.

Dedicated photo cameras should indeed operate in a global shutter manner also with MOS sensor technology. They need to be capable of collecting light over half a minute, even half a hour.

Thus in the field of still photography the rolling shutter issues are essentially limited to smartphones and the like. Video is a different story, though.

Dedicated photo cameras should indeed operate in a global shutter manner also with MOS sensor technology. They need to be capable of collecting light over half a minute, even half a hour.

Well, integrating half a hour directly are capable only CCD's with cryo cooling, the half a minute should be quite common with the CCD's.
However CMOS are not capable of that long shutter times directly (their dark leakage is so high, it would saturate just by that), so a trick is used instead: The shoot is split to multiple shorter (underexposed) shots, each with the shutter time around 10's of ms and then all are summed up in the post processing software in the camera. This then resets the pixels regularly, so their charge stay within the operating range, while the noise is then averaged out by the following processing. With that, the "dark" calibration (to remove the static pixel noise) is applied as well.

In this way you may do integration time in the range of many minutes with quite simple and cheap CMOS sensor, or even hours with CCD's, if the temperature is stable (to have the static noise calibration accurate enough)...