Actor Shipping: A good amount of the fandom tends to ship the pros with either each other or their celebrity partners (if they don't already have significant others), and the show often contributes in adding fuel to the fire. Val is the most common victim of this, being shipped with his Season 16 partner Zendaya ("Valdaya") and Season 19 partner Janel Parrish ("Janelskiy"). Other examples of this are Sharkaroo (Robert Herjavec and Kym Johnson, which did turn out to be Romance on the Set after all) in Season 20 and Switch-Up pairing Alemma (Alek Skarlatos/Emma Slater) in Season 21.
Alemma aside, Season 21 did a decent job in averting this by being the first season to have a married couple competing against each other, as well as featuring plenty of dances dedicated to significant others (such as Tamar's rumba in Week 4) and two pregnancy announcements (Allison in Week 4 and Nick Carter's wife Lauren Kitt in Week 8).
Creator's Pet: While there's no denying he's a talented dancer and choreographer, the constant shilling of Derek Hough (to the point where the judges declared that he and his professional dancer partner deserve to win before the other finalists got their last dance) turned off many fans.
The shilling, plus that he has, thus far, been given reasonable contenders while even the other "wonder triplets", Julianne Hough (his sister) and Mark Ballas (Mark's parents raised Derek and Julianne), have been handed less-than-stellar (or popular) partners.
Zigzagged, Len (the head judge) is not the biggest fan of Derek, sometimes Len's criticism has nothing to do with Derek's partner at all. Nor is he a big fan of Mark Ballas who took creative risks (just listen to him criticizing the zombie paso doble routine.)
During the Hough-less Season 22, we instead get a Network's Pet in the form of Ginger Zee, who got Val as her pro partner and, despite delivering less-than-stellar dances most of the time, receives more praise than the fanbase feels she deserves, just because she's the chief meteorologist on Good Morning America, which tends to report quite a lot on Dancing.
She also received the first perfect score, and made it to the finale, against arguably a much more talented cast. Articles were written questioning whether there was some sort of "fix." 
Elimination Houdini: Marie Osmond, Master P, Michael Irvin, Cloris Leachman, Billy Ray Cyrus, Bristol Palin, Kendra Wilkinson, Hope Solo, Melissa Gilbert, Bill Engvall, Chris Soules, etc.
Season 21 avoided this to a certain degree, with all of the weakest celebs being eliminated during the front half of the competition. Still, a lot of people weren't pleased to see Alek Skarlatos, who among the remaining celebs wasn't exactly the best, in the final three just because of his personal story, much like what happened to Noah Galloway in the previous season.
Ensemble Darkhorse: Sharna Burgess, who is considered one of the better pros and even nicknamed "Lord Sharna" by a good portion of the fanbase, but hasn't scored a mirrorball yet. Troupe member Jenna Johnson could also qualify.
Germans Love David Hasselhoff: To say this show made Bindi Irwin popular in North America would be an understatement. It really is no wonder she was her season's champion.
Growing the Beard: After losing its audience share due to the overblown double night set-up and a general negative reaction to reality shows, the change in format to have only one show a week has paid off, as has getting Erin Andrews to take over the sidekick role from Brooke Burke.
In week 2 of season 20, Nastia Liukin and Derek Hough danced a rumba to "Thinking Out Loud". Season 20 was the season the dancer on the song's music video, Brittany Cherry, made her official debut in the troupe.
During Nick Carter's first dance in Season 21, his pants' button popped out, reminding him of Joey Fatone's microphone coming loose during his first dance in Season 4, which was set to the Bee Gees' "You Should Be Dancing". Guess what song Nick ended up dancing to on Famous Dances Night five weeks later?
Hollywood Pudgy: Former pros Lacey Schwimmer and Cheryl Burke for a time underwent backlash because of their weight, though that obviously didn't hinder their dancing skills.
Ho Yay: Ubiquitous. Lest we not forget Mark Ballas and Val Chmerkovskiy going head to head.
The Madonna dance by Carson Kressley with the Chmerkovskiy brothers is dripping with this. There's no doubt it was intentional.
What likely started it was the season with Steve Guttenberg. One week, his partner, Anna Trebunskaya, had to take a day off. To avoid having him fall behind, her then-husband, Jonathan Roberts, filled in to help Steve. This led to them performing their all-male Tango (which Tom dubbed "The Man-go,") on the results show that week.
Then there was the (in)famous kiss between Bruno and Donny Osmond.
Bethany Mota's samba: Derek Hough and troupe member Sasha Farber's accidental(?) kiss.
Season 21 also had its fair share of homoerotic undertones, for the most part involving, oddly enough, Carlos PenaVega (who's not only married to a woman but was also competing against said woman), such as the footage of him and Val practicing hip action for Carlos' salsa with his actual partner Witney, and an entire montage dedicated to Carlos and Alek's bromance in the finale.
Team Bruno's tango had Jodie Sweetin dancing with Peta Murgatroyd and a shirtless Nyle DiMarco dancing with an also shirtless Keo Motsepe for a bit. As some people have said, that was the most Bruno thing anyone has ever seen.
Just Here for Godzilla: Every season will have someone notable enough to be cast that will encourage viewers to tune in.
Season 17 itself had a quite well known cast for that time, with Snooki, Amber Riley from Glee, and even Bill Nye himself.
One of the most notable ones was the elimination of Sabrina Bryan in Week 6 of Season 5, which shocked and brought tears to everyone. And it happens AGAIN, in the SAME WEEK, to her in the All-Stars season! In a cruel coincidence, Sabrina was eliminated on Oct. 30, 2012...exactly five years to the day she was eliminated the first time!
Despite constantly getting judge praise, almost getting the first perfect score of the season (in the Most Memorable Year week she had a 39), and being one of the best dancers, Willow Shields in Season 20, she was eliminated in the seventh week, to the shock of everyone on the show and most people watching.
The reaction of Hines Ward after he and Kym Johnson completed their Argentine Tango three days after Kym nearly broke her neck.
JR Martinez's rumba to "If You're Reading This". The dance was dedicated to his fellow soldiers who didn't come home from Iraq and Afghanistan. There wasn't a dry eye in the house when the dance was done.
Len Goodman's story on how he can no longer dance due to his knees...then joining in on Nastia Liukin and Derek's Viennese Waltz in his honor.
Nick Carter and Sharna Burgess' Foxtrot to "Comin' Home, Part II", dedicated to Sharna's father and grandmother, during Hometown Glory week.
Their week 8 contemporary dedicated to Nick's wife.
Bindi Irwin's contemporary to "Every Breath You Take" in Most Memorable Year Week. For that matter, the package beforehand counts as this too. There probably wasn't a dry eye in the audience after the package and dance. Turned Up to Eleven after her freestyle to "Footprints in the Sand", when Derek surprised her by having a picture of young Bindi with her father displayed prominently on the wall at the end. Bindi took one look at it and crumpled into tears in Derek's arms.
Speaking of Most Memorable Year week, Lea Thompson's contemporary to "Dance with my Father".
Most Memorable Year Week strikes again in Season 22, with Doug Flutie's tribute dance to his parents (who died literally within an hour of each other after being married for decades), set to "The Rainbow Connection".
Two weeks later, during Switch-Up Week, it was Nyle DiMarco who turned on the waterworks as he and Sharna danced a Viennese Waltz to "I Get to Love You" and scored the first 10 of the season.