The tiny exception
However, Part II of Schedule 2 hides a tiny exception to the above. In this part, it's column 2 that indicates whether the offence is one that could attract points.
The first row is covered as above - a cyclists cannot commit that offence.
The other rows are a bit more tricky - you can commit these offences without driving (without, in fact, having ever sat in a car at all). For example, if you go equipped for stealing a car, you can get 8 points on your licence. (As an aside, an interesting contrast there - kill someone with your car = 4 points, appear to be preparing to steal someone else's car = 8 points, do our law-makers really have their priorities straight?)
The point is, you can get points on your licence if you are cycling equipped for stealing etc. So, in the strict sense of the question at the top of the page, you
can get points on your licence for something done while cycling.
What if you don't have a licence
It's sometimes said that cycling offences can't attract points because a cyclist might not have a licence to accumulate the points. Although the conclusion si right, the reasoning is baloney - there's a mechanism by which the points accumulate even if you don't have a licence. When you get a licence, it will have the points on. It's the way under-age joyriders (for example) get points. It's set out in
Section 45 of the Road Traffic Offenders Act, which has such phrasing as "whether he is at the time the holder of a licence or not" and so on.