Woah, folks!!!! Too many bad terms being thrown around here.
First, safety stops are recommendations. Second, some people keep calling these a "deco -stop"...quit doing that!!! If you are doing recreational diving, your are NOT doing "deco" diving. Wrong forum, wrong place. All of the charts being referenced are for "no decompression diving", which is staying within those chart limits. Again, NOT DECO!!!!
Okay, so what "should" you do? First, understand the purpose of the safety stop. Anyone remember what was taught to them in their OWD course? The safety stop is an "added safety measure of 3 to 5 minutes at 10 to 30 feet that divers should take after no-decompression dives to help reduce silent bubbles and the risk of pressure related injuries." (reference SSI manual).
Okay, first understand that the safety stop is a recommendation. Does it mean you won't get bent? No. There are too many factors involving DCS. The whole purpose of the recommended safety stop is to let the blood circulate throughout your entire body, which will help get rid of the very tiny bubbles that may (or may not) still be present in portions of your body. The idea of 3 to 5 minutes was to allow these bubbles more "contact" time with the blood to go back into solution, which will then be off-gassed through the normal respiratory cycle.
I have no idea as to going back down if you blew your stop. Would I do it? Yes, I would, with the simple understanding that if those bubbles were still there, then the increased pressure would help drive them back into solution in the bloodstream, rather than the bubbles being transported to an area of the body where they could do harm. Again, they would be off-gassed through normal respiration (with varying times, depending on a lot of factors, below). Please understand the physics of what is happening with the air bubbles, and that going back down to a safety stop depth does nothing detrimental - at all. If you have ever watched your computer during this time (it's interesting to see how they react), they often will increase the amount of time for the recommended stop (I dive Suunto's, and that's what mine tells me).
But to say that if your computer didn't "tell you" anything was wrong is a misunderstanding of the gas laws, and how the lack of a safety stop could (note the would "could") impact you. There are so many variables involved (how much residual nitrogen you have, how many repetive dives have you already done, how hydrated you are, dive depth, etc.), that it isn't something to trifle with. Computers, for the most part, have conservative dive profiles. But computers are no guarantee how an individual's own physiology will specifically impact them. Again, the safety stop is only a recommendation, not a requirement.
I have done wall dives, where we came back up so gradually (working back and forth along the face), that the safety stop was "built in" to the dive, and while the computer "triggered" the safety stop when I hit 20 feet, I just kept on swimming.