Many of his concerts have been televised, of course-- notably his two Central Park concerts, with Garfunkel and solo-- and he has appeared to accept various awards and inductions, sometimes performing as well. He has been profiled in shows that, well, profile celebrities, one per episode... and even one called Album Flash that highlighted one new album per episode; Simon's was Hearts and Bones (it's on YouTube).
It goes without saying that has been part of many shows premised on the "100 Greatest" songs of a given era, songs from soundtracks, etc. And in 1993, he was part of a mini-series, Rhythm and Jam, aimed at teaching the basics of music to children (other guests included En Vogue and the Red Hot Chili Peppers!).
Other shows follow the American Bandstand format, with appearances by several bands over the course of a show. In the 1960s, Simon and Garfunkel appeared on such shows, called Stramash and Shivaree. In 1980, it was The Midnight Special, which featured concerts by mostly acoustic acts. Also in 1980, he co-hosted (with Andy Gibb) Solid Gold, a disco-dance show.
Then in 1987, Simon performed on the "Gospel Session" of Cimeax Sessions, which usually featured one performer in concert per episode, often a jazz great. Another, 1981's Music Central, never flew, but the interviews and performances (by Pat Benatar, Blondie, and mostly other women) were salvaged and repackaged as a one-hour special.
Simon has appeared on television to pay tribute to fellow musicians like Stevie Wonder, Brian Wilson, Willie Nelson, and Harry Belafonte, including some he has worked with extensively-- such as the Dixie Hummingbirds and Ladysmith Black Mambazo-- plus comedy friends Lorne Michaels and Steve Martin, and even Muhammad Ali!
And he has been part of historic commemorations for-- and fundraisers for-- 9/11, Hurricane Katrina, autism, Kurdish refugees, and Vaclav Havel's democratic work in Czechoslovakia. Also partiotically, Simon (a lifelong baseball fan) has sung "The Star Spangled Banner" during the World Series (1986, Game 6).
Most notably, Simon is the only musician to be in the "Five Timers Club," that list of people who have hosted Saturday Night Live five times or more. The show's entire second episode, back in 1975, was pretty much a Paul Simon special. In fact, a montage of his SNL appearances was part of Words and Music, the special exhibit on Paul Simon at the Rock and Roll Hall of Fame. In all, Simon has been on the show 13 times, which makes listing these appearances unwieldy. (He was also in Lorne Michael's other sketch show, The New Show, which only ran one season; he played Abraham Lincoln, which might explain that.)
However, it is important to note that Simon has had more substantial appearances on several iconic television shows, for some of which he was able to perform more than one song:
1957 American Bandstand: "Hey, Schoolgirl"
1964 Ready Steady Go: "I Am a Rock"
1966 The Ed Sullivan Show: "I Am a Rock"
1966 Hullabaloo: "Homeward Bound"
1967 The Smothers Brothers Comedy Hour: "Cloudy," "Homeward Bound," "Overs" (two episodes in one year)
1967 Paul Simon and Art Garfunkel: Little is known about this British special on the duo.
1968 Kraft Music Hall: Trio for Tonight (the "third" was Nancy Wilson): 10 Songs, mostly from Parsley Sage
1969 The Bell Telephone Hour/Songs for America: "America," "So Long, Frank Lloyd Wright," "Bridge Over troubled Water," "At the Zoo," "Scarborough Fair," "El Condor Pasa," "Punky's Dilemma," "Mrs. Robinson," "Feelin' Groovy," "Mystery Train" (Elvis cover), "For Emily, "The Boxer," "Homeward Bound," America," "Sound of Silence," "Song for the Asking."
1977 The Paul Simon Special: "Bridge...," "Still Crazy...," "Old Friends" (with Garfunkel), "Loves Me Like a Rock," "The Boxer," "Slip Slidin' Away," "Something So Right," "I Do It for Your Love."
1977 Sesame Street: "Me and Julio..." "El Condor Pasa" In later episodes, Simon would duet on the children's song "BINGO" with his son, Harper, still a child, as they demonstrated how vinyl records are pressed. He also sang the alphabet with Ladysmith Black Mambazo on the show (they have made several appearances of their own without Simon), and cameo'd in the all-star version of "Put Down the Duckie." These came in 1989, 1990, and 1992.
1980 The Muppet Show: "Scarborough Fair," "El Condor Pasa" (Gonzo performs), "Long Long Day," "Fifty Ways..." (Electric Mayhem sings), "Baby Driver" (also by puppets), "Loves Me Like a Rock," accompanied by Electric Mayhem
1993 MTV Unplugged: "Born at the Right Time," "Me and Julio...," "Graceland," "Still Crazy...," "Mrs. Robinson," "Bridge...," "Something So Right," "Boy in the Bubble," "Late in the Evening," "Homeward Bound."
2008 and 2010 The Colbert Report: "American Tune" and "Getting Ready for Christmas Day," respectively.
2011 The Harmony Game: songs from the Bridge... album, as this is a documentary about that work.
2016 Simon sings "Bridge over Troubled Water" at the Democratic National Convention in Philadelphia, at which Hillary Clinton is nominated for US President.
Some compilations of Simon on TV-- mostly with Garfunkel-- are available on Amazon. I have:
-Simon & Garfunkel: Broadcasting Live
-Simon & Garfunkel: See for Miles 1996-2004 (contains the Songs for America special. No idea what the Who reference in the title is for!)
-Simon & Garfunkel: Across the Airwaves