There can be many factors in play here so it does pay to do some checking.
First potential cause, especially with reloads, is primers that aren't fully seated. The check is to strike that round a second time using Double Action. If that round then fires it's fairly likely the primer wasn't fully seated and the first strike acted to seat the primer completely. BTW the reason you test for this using Double Action is the shorter hammer fall transfers LESS energy to the primer.
The second most likely cause is a strain screw that has worn "shorter" or or been shortened to lighten the DA trigger weight. Test for this is to measure the weight of the DA trigger. On revolvers of the vintage of your 15-3 the factory had a target weight for the DA trigger of 9.0 lbs. +/- 1/2 lbs. If you find the DA trigger weight is under 8 1/2 lbs. then I would suggest replacing the strain screw with one of the correct length for your vintage revolver. BTW, Round or Square butt are different lengths and early 70's Round butt are a different length than those for the current round butt K and L frames. (found that out with my 19-3, a spare from my stock of part for modern K and L frames wasn't close to being long enough).
The third most likely potential cause is a mainspring that was re-profiled to lighten the DA trigger pull or is worn out. Test for this is to check the DA trigger weight with a NEW and CORRECT strain screw installed. If the weight is still lighter than the target then a new mainspring is the solution. BTW, I've never compared early 70's mainsprings to late model ones but everything I've seen on the Net and in catalogs indicates they are the same.
Fifth most likely cause is a lockwork that is really gummed up. Back in the 70's a common method for cleaning and lubing the lockwork on a S&W revolver was to hose it out with WD-40. A rust preventative oil that takes on some characteristics similar to glue after 15 or 20 years. The solution is a complete removal of all the internal parts and a long soak in mineral spirits, a good scrubbing, a rinse in alcohol, and then re-assembly with the judicious applications of tiny drops of Mobile One motor oil at the pivot pins, rebound slide, and cylinder stop.
Sixth potential cause is a "short" hammer nose or a damaged tip on the hammer nose. BTW, the Hammer Nose is the part on the hammer that most would call the firing pin. Test is to open the cylinder, pull the cylinder release button to the rear and cycling the action to bring the hammer to full Fall with the trigger held to the rear to keep it all the way forward. Then lay a dime next to the tip of the hammer nose and look to see if the tip is even or slightly extended past that Dime. If it's short the solution is the installation of a new Hammer Nose. BTW, the Power Custom Extended Hammer Nose available at Brownells is well made with superb installation and fitting instructions and it WILL fit a 15-3. Know that because I had this particular issue with a 1972 vintage 19-3 that shipped with a short hammer nose and was nearly unfired when I found it at a gun show a couple of years back. That 19-3 had a nose so short that it wouldn't ignite a Magnum primer even in single action and failed to fire about 20% of the time with a non Magnum primer in single action. It only came to about 1/2 the thickness of a dime.
