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.
Family-Unfriendly Aesop: There are some instances which may give the false impression that cheating is okay. In "A Splash Party, Please," when Barney and the kids are having a tug o' war, Min helps the other kids win by tickling Barney. Later, in "Falling For Autumn," Shawn participates in a relay race with a peanut stuck to his spoon with peanut butter. Proponents states that it's safe to assume that these "cheating" ways were just thrown in as jokes (and the characters playing them off as such), while opponents state that children of the target demographic pick up from mimicking and may copy the action because they do not understand that it's supposed to be a joke (Sadly, in some countries - particularly corrupt ones, this Aesop is Truth in Television).
What's next? "It's okay to throw barrels of oil into the ocean"? ...Wait...
Moe: Tina (in the early Barney & the Backyard Gang videos), Kathy (in Season 1), and some other kids. One girl, Linda, seems to make a connection with Kathy in a reunion special.
Narm: Sandy Duncan singing "I Love You" at the top of her lungs, when she's by herself in an early video.
One could argue the acting of the early child cast members is pure narm as well.
Nightmare Fuel: Depending on who you are, the post-credits during the 90s could have really gotten to you. There was the 1993 Connecticut Public Television logo, the old Corporation for Public Broadcasting/Viewers Like You funding credit combination (which was originally accompanied by a voiceover given by a cast member and spoken so fast it could be scary), and finally, the 1993 PBS ident featuring the creepy singing P-Pals which closed nearly every children's program on the network until PBS Kids debuted in 1999. Could potentially be a triple scare factor.
To some, the original Barney costume from the Backyard Gang videos looked pretty creepy.
Nostalgia Filter: Among others, this is one reason that the show has fans older than the target preschool audience.
Older Than They Think: The "I Love You" song first appeared in the Backyard Gang videos, but the song was written in a 1983 children's book prior to its usage on Barney. In 1992, the show's producers made an agreement with songwriter Lee Bernstein where she'd get credit for the song in future episodes (and was paid $6,000 for use of the song). This didn't stop said book's publisher from suing both Bernstein and the produers of Barney & Friends in 1994 over the song's lyrics. The lawsuit was settled later that year and "I Love You" has remained part of the series.
Painful Rhyme: This oddity from the original "Backyard Gang" videos. A rare example of the "painful rhyme" preceding the desired word:
Jason is in Barney's club, and Adam has a ticket; Our mom has never seen him, 'cause she doesn't know our secret.
Special Effect Failure: In the original three Barney & The Backyard Gang videos, Barney would close each video winking (with a crudely drawn eyelid). In later videos, he would wink with a sparkle in his eye, which continued in the television series.
There was also an attempt to animate a breakdancing Barney for the first usage of "Mr. Knickerbocker," using stop-motion animation.
Tastes Like Diabetes: One of the largest targets of this criticism... There are bullies in real schools, dammit!
Unfortunate Implications: During the song "The Wheels on the Bus" in "Round and Round We Go", the black girl and the Asian boy are in the back, with the two white kids in the front.
The show's other songs in the same production became Woolseyisms, though ones mostly true to the originals. One of the more dramatic examples "I've Been Working on the Railroad", which now talks about working at a sea port and watching ships. Another example noted on the main page is "London Bridge," which became "Yarkon Bridge" to connect with Israeli audiences.