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.
In spades with all the members, but Jonghyun and Key deserve special note, to the extent that Key usually refers to Jonghyun as "honey". There's a YouTube channel dedicated entirely to video clips of them pumping up the gay together, like sharing Pepero.
Minho and Taemin definitely as well. They have a very brotherly relationship and are often seen hugging or messing around together.
Outside of just the other members, Key has plenty with best friend and (as of SM's merging with Woollim) labelmate, Woohyun. Even other idols comment on the way they act like a couple - once when Woohyun was accused of texting his girlfriend while brushing his teeth he replied, "it's not my girlfriend, it's Key!"
Dispatch tried to catch Taemin on a date but instead he was hanging out with fellow labelmate and friend Kai of EXO, who collaborated with him on his solo mini album. This launched one of SM Entertaiment's biggest ships to date: TaeKai.
Internet Backdraft: A small but major one: In 2013, at SM Town Week, Onew and Minho donned Brownface with mustaches, poncho tunics, and sombreros (fake Rick James dreads for Minho) and danced to "Macarena". On one side, there were people who were shaking their head in shame and on the other side, there were angry fans screaming "OPPA DIDN'T MEAN IT".
Key's and Minho's pants look a little...weird and out of place during their 2011 K-Pop Music Fest part in Australia.
Taemin's pencil leggings in "Dream Girl"
Yoko Oh No: It was confirmed in October 2010 that Jonghyun had been dating a popular actress, Shin Sekyung. This was the general reaction from both his fangirls towards her and her fanboys towards him. However, it seems like many fans may have actually accepted the relationship because SHINee's fanclub numbers have been on the rise. Unfortunately the couple broke up due to busy schedules.