I will start from the beginning to mention that "Supernatural" can be considered one of the best TV Shows in the history of television (I am saying one of the best only to be politically correct because in the fandom's heart this is truly the best one); however, in over 250 episodes there are bound to be some episodes, story lines, and plot holes that don't quite match the tone of the show, or just were not portrayed very well.
Sam and Dean fighting with each other for no apparent reason
There were truly heart breaking moments when Sam and Dean had to lie to protect a life or for the good of everybody, but keeping secrets, or telling a deformed truth was by all means unnecessary.
There are a lot of episodes, and story lines based solely on the brothers being annoyed at each other and constantly bickering. It is understandable when there exists a big issue that needs to be resolved, but when they argue all season, and only make up during the finale, it does not make much sense. For example, the conflict after Dean let Ezekiel possess Sam's body could have been handled differently.
How Mary acted towards her sons after being brought to life
We were all very excited to see all the Winchesters (almost all) to be back together as a family, but Mary did not turn out to be the mom the fandom expected. It was understandable that she would have difficulty adapting to the world three decades after her sudden death; however, how she treated her sons was unacceptable.
During 11 seasons, Mary Winchester, although was not portrayed as a perfect mother, but she should have shown more compassion and love for the boys standing in front of her. She lied to them, worked with their enemy, and frankly ignored their attempts to reconnect. But she did sacrifice herself in the end and that is definitely a "hunter motherly" love.
Dean going dark side
It was interesting to see Dean becoming "supernatural" too in some sort, because we were only used to Sam Winchester's psychic abilities; however, the writers should have handled it better. Seeing him struggle and complain all the time became tiring for the fans, and it also caused major conflicts between the brothers (brotherly love is the essence of the show after all).
Castiel's unnecessary death
By now, all of us have been used to the unnecessary deaths in 'Supernatural'; however, the death of Castiel in the finale of season 12 was just wrong. Firstly, all the fans knew that he would somehow make a comeback (which he will in the first couple of episodes in season 13), and secondly, the number of deaths in a single episode was too damn high. We had the death of Crowley, Mary, and Castiel, and although only the self-sacrifice of Crowley will stick, it made the rest of the deaths seem just crowded.
And now for some honorable or rather dishonorable mentions are when we have the whole story arch with the Leviathans, how Charlie died, and how Sam's love interests kept dying over and over (Sam Winchester's love interests have a death rate of 86%).
Sam giving up hunting
Although Sam is a bookworm and truly wanted to have an apple-pie life in the the first seasons of the show, it became obvious that that life would remain simply a dream. In the many occasions that Dean died, Sam could barely function without his brother and did everything he could to bring him back (including making demon deals and putting his own life in danger).
However, the writers decided to make a bad plot twist, by having Sam move on with his life completely (with Amelia, whom the fandom could not even stand). It is not that we would not want to see Sam happy, but the way the writers did it seemed completely out of character, and just forced. All in all, it sucked.