There most certainly is a stigma attached to serving on a jury that hangs. (I won't even analyze the origin of the phrase "hung jury" to prove my point.)

When you're summoned to jury duty, during voir dire the judge will ask you whether you ever served on a jury before and, if so, whether you reached a verdict. What she means when she asks this is whether you and your eleven colleagues were able to reach a consensus. If not, you are not a good candidate for a jury in her courtroom.

Once a hung juror, always a hung juror. It's a privilege to serve on a jury, and, if you waste the court's time once, you aren't likely ever to have a chance to do so again.

If the judge doesn't dismiss you from the jury panel, the prosecution most certainly will. The only lawyers who like hung juries are defense lawyers.

I might add that the media also view hung juries as failures. The Phil Spector trial coverage proves this. After the trial, the jurors were interviewed on camera and all the commentators clearly sided with the jurors who voted guilty (ten of twelve). The two jurors who felt there was reasonable doubt were characterized extremely unfavorably by the press.