These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Awesome Music: Most of the background music was improvised live during taping by jazz pianist Johnny Costa. Refusing to play "kiddie music," Costa filled his accompaniments with sophisticated runs and flourishes. His arrangements were often compared to the legendary pianist Art Tatum, which is about the biggest compliment a jazz pianist can get. In fact, Costa was given the nickname "the white Tatum"... by Art Tatum himself. Now that's awesome music.
Ear Worm: Just try to get the theme song out of your head after reading this page...
The ending theme is even catchier "It's such a good feeling to know you're alive, such a happy feeling you're growing inside..."
Fridge Brilliance: Many episodes open with a shot of a flashing yellow traffic light that was in Mr Rogers' house before panning over to the door where he entered. A yellow light (generally) means "slow down" which is exactly what he did with the pacing of his show, especially compared to others at the time.
Full-body costumed characters like Bob Dog and Purple Panda were known to scare some viewers.
Some viewers also had a childhood fear of the music that played when the Episode Code Number was shown at the end of older episodes. It didn't help that the oft-feared 1971 PBSVanity Plate used to directly follow this.
Suspiciously Similar Song: Some songs used in the early days were co-written with Josie Carey, dating back to their time working together on The Children's Corner. Rogers and Carey sold the copyrights to these songs (foolishly, by Rogers' own admission), and mostly stopped using them in order to avoid paying royalties, as Rogers said that he could come up with new ones that were just as good. Indeed, some of the newer songs have very similar lyrics - for example, "I Like You As You Are" gave way to "It's You I Like".
Fred Rogers liked Eddie Murphy's parody of his own show, "Mr. Robinson's Neighborhood." To be fair, it was an Affectionate Parody, not deliberate and malicious mockery. Apparently, Eddie Murphy caught up with Mr. Rogers and told him, "You understand, we only do it because we love you."
Johnny Carson also did a parody sketch where Mr. Rogers explained where babies come from, but it was also an Affectionate Parody and unlikely any kids were up late enough to watch it.
On the other hand, in the late eighties when Burger King made an ad with a fake Mister Rogers explaining why BK burgers were better than McDonald's burgers, Fred Rogers said "You will stop that immediately!"... and they did. This was mainly because he looked too much like the real him, and did not want kids to get confused. By comparison, obviously no kid is going to confuse Murphy for Rogers, and Murphy's sketch was broadcast safely out of the way of any typical child's viewing time.
Obviously, the Westboro Baptist Church did the exact same thing, albeit in the most Evil Cannot Comprehend Good way possible; they protested his funeral because he was a tolerant person and didn't speak against homosexuality. No seriously, that is verbatim what their beef with him was.