You have to go to the game you played against them (or one of them) and choose Send Trophy from that. The link is on the game information screen.
Can't find old trophy icons

Yes, I know. This is where I could not find any of the old trophies I described. All I found were a bunch of new ones, many that did not seem related to chess (ie re golf, football, baseball, etc). What happened to the trophies that complimented a particular chess skill as I mentioned above?
PS If you mean GIVE TROPHY, this is where I'm looking, if there is another link called SEND TROPHY, I can't find it. I click on my opponent's handle, I get a row of symbols and one of them is a trophy. I click on that and get GIVE TROPHY.
Thanks.

You can get to what you want a couple of ways. The easiest is probably to go to the player's profile. Find your record against them.
Click that link. Find the game you are interested in and go to it. On the information tab, click Send Trophy.

Martin, we seem to keep miscommunicating. Have you looked at the trophies that are shown when clicking on SEND or GIVE a trophy. I get four or five pages of trophies that don't have much to do with chess much less a particular area such as knight or rook play, or pawn play. If you have the time, please look at my trophy case and you will find what I am talking about. I'm really just asking why these old trophies are (apparently) gone and replaced with trophies for football, golf, etc.

Yes, I have looked. Here is a small selection of the options under the Send Trophy link in the game archives.
Most of them are very game specific and include most, if not all of the ones you mention in your first post.

Martin; Thanks for your patience. I finally found them. I was trying to send a trophy during a game and that's where I was having my troubles. Guess if I want to send one of the "old" trophies, I'll have to wait til the end of the game. Any guess why the "old" trophies don't show up when clicking on GIVE TROPHY?
Thanks again for being so patient with an old man.
One of my current opponents played some mighty fine knight moves and I wanted to award him a trophy. I went to select a trophy and could only find silly ones. There used to be trophies for Knight Verturoso, Lord of the Rook, Pawn Magician, Great Endgame, etc. If they still exist, would someone please tell me how to find them. If they no longer exist, sad on Chess.com.