What are these shoots and how can I get rid of them?

I had this in a garden in Essex and one weekend went down at least a meter and the root was still going strong. I removed it at that level and it reappeared the next spring.
Given that it's ancestors have been around since the Carboniferous period you could have your work cut out killing it off.
Best bet is just to remove the growth whenever you see it.
The good news is it has been used as a pan scourer and you can make a beneficial tea out of it.

 

The equisetum plants that I've seen have come up with, for want of a better horticultural term, a knob on the end. They also have clearly defined sections like a bamboo and the most pugnacious weedy type, comes out brown and then greens up.

In the pictures, at least to my eyes, i don't see any of those features but do see a very close resemblance to spring bulbs growth which is happening now. I may well be wrong and it might be equisetum but I wouldn't worry about it until it's confirmed with some better close up pictures.

 

The ones that, like a wizards staff, have a knob on the end are the fertile fronds that come up in spring, after that it's the sterile taller fronds for the rest of the year.
Agree about better photos preferable when a bit bigger.

 

Equisetum has two types of growth, the first in spring, as you describe, which are structures containing spores, and the second in summer when green shoots emerge. It may be too late for these shoots to emerge now but I thought that glyphosate would destroy the young emerging shoots of spring bulbs. I hope I’m wrong though! A closer photo would definitely help.