I've never been on JSA, because up until now I've literally gone straight from school > college > uni without a break so haven't needed it.
I don't think that people should feel embarrassed, because it's there to support you while you're searching for a job. However, the number of people who abuse the system means that everyone gets lumped in with them, and the "general public" seem to assume that being on JSA is something that you should be ashamed of, that you're lazy, that you have no intention to work etc. You've heard all of this countless times most likely.
I have several friends who didn't go to college when the rest of us did, which was five years ago, and they just went straight on JSA and have stayed on it since. They get enough to pay for a room in a shared house which they have together, and enough spare to feed themselves and have a better social life than I have ever been able to afford even though I'm now working. They apply for the bare minimum jobs, but make sure that there's no way they'd get an offer by handing in a poor CV.
How do I know all of this? Because they BRAG about it. One of them has even managed to fabricate a disability (back problem) to mean that they can't work, so he gets disability as well. Absolutely nothing wrong with him.
On the other hand, my Dad, who is 51 and has worked full time without a break since he was 16, was made redundant in November. He was on JSA for a period of just one month before getting his new job (which he was completely overqualified to do, but he would rather do ANYTHING that NOTHING- and has now almost finished training to be a driving instructor) and applied for HUNDREDS of jobs within the two week period of leaving his job. He got a fair number of interviews because he pushed SO hard to try and find work. He attended his Job Centre meetings in a suit and tie, because to him, that's how you behave in what is supposed to be a professional environment. In the town I live, there are mostly hoodies smelling of weed in our Job Centre, so imagine how he felt in his suit..!
Nobody should be embarrassed to ask for help when they need it, but unfortunately due to a number of idiots ABUSING the system, JSA is given a bad name.