The simplest and most effective way to hold clay targets for target shooting is to use the target hangers made by Sure Site (SureSiteInc.com), or at least that's what I've decided after trying a handful of other methods, and then finally trying these out.
Clays are a blast to shoot (pun intended), they're inexpensive, they don't harm the environment, and they have the advantage of giving the shooter immediate feedback. Also, you can use them for target practice with all types of guns. The inexpensive fun associated with clays tells me you can't go wrong making them a regular part of your shooting excursions. Clays become incredibly versatile when you can hang them to shoot them, and Sure SIte target hangers make that possible.
At just $5 per 10 pack, I absolutely recommend these plastic hangers made by Sure Site. They're not indestructible, and they can get lost in the rubble (a problem you could easily solve by spray-painting them a hi-viz color), so you might want to buy a 50 pack and have some REAL fun. Shipping isn't the cheapest, but perhaps you can avoid that by getting your local sporting goods store to carry them. In the mean time, order as many as it takes to make the shipping cost worth it to you - you won't regret it.
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https://www.youtube.com/watch?v=4EIbtTa6HbU