I got stop for speeding in United States (New York City) and I was issued a summons stating my offence. I was than told to mail it back to the court, stating if I plead guilty or not. I did so when I was in Las Vegas and mail it through the concierge at Monte Carlo. I came back to Singapore and after almost a year, I have yet receive my summons. I was told by my friends who were caught for speeding that they received their summons after 6 months. After reading some forums, I heard that if I fail to pay the fine, and return to the US, I might get arrested at the airport. I will have to pay the fine with interest. I have the intention of returning to the States, and I'm worried.
Simplest answer to all your questions: If you're worried pay the fine and forget this ever happened. NY state is not likely to pass along traffic violation information to Singapore.
So with that in mind: you could try and go to NYS Traffic Violations Page and pay your ticket, unless of course you decided to plead "Not guilty" or your violation required a mandatory court appearance.
For New York City this may be different but the back of your ticket should have information on how to plead and pay your ticket online. Most forms are available from NYC.gov (though for some reason seems to be broken today).
If you mailed in your ticket with a guilty plea and you expect back documentation on what the amount of the fine should be, you shouldn't have anything to worry about, whenever this arrives.
Unfortunately if you didn't pay it on the web and mailed a check, there's no confirmation that you paid it. Just rest assured knowing that you paid it.
Besides, moving violations like speeding tickets aren't offenses people get arrested for by themselves. US border agents won't arrest you unless there is an outstanding warrant for your arrest, an unlikely scenario unless you committed a crime while speeding. However if you have an outstanding speeding ticket, you will be subject to additional screening upon re-entry to the US. Further information can be obtained here.Good luck and sleep a little better knowing you won't be arrested upon re-entry if they didn't receive your letter ;-)
Since you're expecting a summons, I'm guessing you pleaded not guilty? Wouldn't it be more expensive to show up for court in the US then just paying the ticket?