Can you eat them(not the trophy sized ones, of course, but the smaller ones)?
I literally know nothing about it...
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I'll answer this part first since it is easiest. In many cultures they are eaten regularly, or at the very least as part of holiday tradition.
The Polish for instance eat them as a Christmas fish or Holiday carp..where markets will open weeks before the season to sell whole/live carp out of large containers. The fish markets will sell them live or butcher them on site for the buyer to take home.
Gefilte Fish can be found in most grocery stores in the Jewish or Hebrew sections. Mainly a mix of white fish/pike/carp etc depending on taste preference.
The problem of course is quality of water dictates quality of fish and carp, being very hardy, can survive and thrive in bad conditions..thusly giving them health warnings as well, and a bad taste. Carp are, from clean waters and prepared properly not a bad tasting fish at all and were originally brought to this country to serve as such due to mankind decimating native fisheries.
I have eaten them, in my childhood and did not mind them at all. Though I am strictly catch and release now, I did post a recipe or two here and will add if you prepare them, remove the dark fatty sections and beware the "floating" ribs they have developed to support their size.
so how does one go about specifically targeting carp?
people used to snag em back home in Oklahoma... there they're not technically considered a 'sport fish', so there's no size/creel limit on em. I've caught one or two many years ago, but just by accident.
This is a much harder question. Answers will come in many forms and really targeting big fish is not just a matter of sitting down and casting a line. People will catch carp from time to time as they are curious and in some waters have come to grow accustomed to human presence. Though the more you learn the more you will catch, the bigger you will catch and so on. If you are really interested check out the carp forums here, maybe stop by one of the events we host through out the year or check out some of the great websites out there and of course hit me up or start a post with questions and maybe we can help show you the joys of carp fishing.
As far as snagging them goes. No, they are not "considered a sport fish" by most, but this is also due to years of people scapegoating them for many of the issues we ourselves caused. Combine to this the new "Asian Carp" threat and bias against all carp has again risen. The problem in many places is not the common carp, but man's lack of understanding or caring about the world around them.