I am getting bamboo shoots (I think) growing through my lawn. I have bamboo close by so am assuming this is what it is. Please can you advise?
Yes, bamboo can absolutely grow in grass, and if one of your neighbors has it, there's a good chance that's what it is. It can be extremely invasive, so if you're wishing to get rid of it, your only two choices are to treat it like grass and mow it, or dig it out (Continuously. For 2-3 years.). Bamboo is almost completely immune to herbicides unless you treat it when it's still very young, and treat it frequently.
If you want to prevent the further spread of the bamboo from a neighboring area, these articles will help you: