I know I will get killed for this, BUT --
I use a Lesche 38D "King of Spades" (i.e. the Ground Shark 38) digger anywhere I can get away with it. I know that to some, using this kind of digger in a public place is "anathema," and I'm sure I will get called names for admitting this. But, that's OK. I do it for a reason -- I feel I can dig the best, cleanest plug possible with this digger, and replace it with the least amount of damage to the turf, and leaving nearly no trace that a hole was ever dug.
I know that some folks think that being seen with such a digger in a public place, that people would freak out and the spots would be "shut down" immediately. I'm sure that is true in some places. In the places I hunt though (here in Oklahoma, and in western PA), I've never had a problem. I am fully prepared to explain the "why" as to my choice to use this digger, should anyone ever confront me on it; I like to think that after I show that my reason is actually to KEEP from doing turf damage, not to "tear up the park," and possibly demonstrate how I cut a plug, that it would "defuse" any issues. So far though, I've never HAD an issue...which says something to me.
Now, with that said, I DO try and "feel the place out" a bit; I don't use this digger EVERYWHERE. I exercise at least SOME common sense, and try to gauge the area I'm in, the possible "attitude" of the neighborhood, etc. Dirt type matters, also. Wet, clayish dirt allows you to dig a GREAT plug with that type of shovel. Dry, crumbly dirt, not so much. So, I alter technique/tools accordingly. But, overall, I still fall back to the idea that in my mind, look at how they cut a new "cup" on a golf course green. They don't do it with a hand-held digger, they do it with a large, rounded "cutter." That's what I try to emulate with my 38D...
That said -- to answer the original question, my plugs are usually circular and about 5-6" in diameter...and usually deeper than the target. I try to dig deeper plugs than necessary (at least, for shallower targets); I try to go at least 6" to 8" deep, to keep as much root system intact as possible. If my target is in the plug, I simply crack it in half with my hands, retrieve the target, put the plug "back together," and then drop it back in the hole. It works well for me.
Steve