Elliot's wedding ring was probably the most known and constant prop on the show. He had worn his ring constantly (very rarely had it off) until Kathy left him. But the ring remained on until the middle of season 7; halfway through the episode (Name) he had it on and then it was off. It remained off until the beginning of season 10. It was a season (9) after Elliot and Kathy already reunited that the wedding ring was off. Why? There was a lot of speculation and curiosity about it during that time, especially with the Elliot-Olivia shippers; even a few non-shippers who weren't fond of Kathy or even liked her had questioned it. There were a few theories: men aren't really jewelry-maniacs as women are. There are happily married men who prefer not to wear a ring or would wear it from time to time. There's also weight gain where it would get to the point where the ring wouldn't fit and cause a problem with blood circulation. The Elliot-Olivia shippers took it as a sign that the Stabler marriage was in trouble. Chris was asked about it on his official site (it wasn't just my question, there were a few related questions from others as well) during that time period- he responded by thinking it wasn't a big deal. The ring 'prop' was Chris's real-life wedding ring and he doesn't wear it as often so it may have leaked into his character. As mentioned before, the ring returned and remained put until the end of season 12 when Chris/Elliot left the show. The only time Elliot had his ring off was when he went undercover as a single man.
As for Kathy, she had hers on until she left him in season 6; although some viewers claimed to have seen it on in Blood. But in season 9, after Elliot returned home and she was pregnant, Kathy's ring returned- but it was silver- probably as a prop as Isabel wasn't married at the time. Isabel did remarry, but her real-life wedding ring was silver while her character's ring was gold (from seasons 10-12 looked like from seasons 1-5). Her season 9 wedding ring could've been a bigger ring for Kathy's pregnancy/weight gain...yet Isabel wasn't pregnant.