Although there is no current draft in the US, registration at age 18 is required. It is not optional for males.
Local draft boards do mot exist as they did during the Viet Nam era.
However, Regional Draft Appeals Boards still exist and are staffed by volunteers in case a draft is implemented.
Both myself and another local Yid in my small CT town serve on the Regional Selective Service Appeals Board for Southern New England. We receive training materials several times each year.
Students looking for Financial Aid for Colleges and grad schools and filling out the FAFSA form are asked if they have registered. If they have not, they are not eligible for aid, grants, subsidized and unsubsidized federal loans, etc.
As for the OP’s question: “Do you register your sons?” the answer is NO, Your sons must register themselves when turning 18.