You have a common problem, but you have to take action soon to prevent an accident. I have used those type of trace clips for years with no problems, but they can get bent up so that they don't do a proper job of holding your traces on. First, are you sure you're putting the traces on the singletree properly. The slot in the trace goes over the single tree, and the clip holds the trace in place. Don't laugh, I've had some folks lay the trace flat on the singletree and put the triangle of the clip in the slot. Honest, it happens.
You can do some things easily to make sure your traces stay on. The easiest is to tie a short piece of leather lace around the singletree, passing it through the triangle on the clip and tieing it tight. Easy and safe. You can use a plastic cable tie in the same way, but you have to cut it to remove it and use a new one next time,
You can also get different style trace clips and replace what you have. Look in a carriage supply catalog and you'll see the different types. Iowa Valley Carriage has them and they just fasten on the singletree in place of the clips you have.
You can also change your single tree in the type that comes with the trace holders on it. You'll see this on higher priced carriages, there is a brass "T" shaped end on the trace and your trace slot fits over the "T" to hold it on.
Look at some carriage supply websites and it will be obvious. Just get it done, because the last thing you want to do is have one trace come loose. Good Luck