Barry's abduction scene, the general scenes of electronics going haywire, and to many, Roy's obvious descent into alien-induced madness. It's just so very deadpan and the setting so banal, yet his madness is profound.
Jillian's attempt to phone for help being frustrated by the aliens having tapped into her land line. The repeating five tones create the effect of a Madness Mantra while also driving home just how helpless and alone Jillian really is.
Johnny Mathis singing "Chances Are" is just the cherry on the icing.
According to the making-of documentary, Barry's actor wasn't afraid of the abduction scene at the time. Not only was he 'abducted' into his real-life mother's arms, but Spielberg had told him all about what was going to happen a few hours before.
The scene where the government gases the escapees trying to climb Devil's Tower is very disturbing.
And approaches Fridge Horror when you consider that the order to deploy the gas was issued with no cautionary warning to avoid dosing the trio with it when the climbers were, say, halfway up a high, sheer cliff.
The scene where Roy is in his truck and has an alien encounter - especially when there's a bait and switch when you think the truck's lights are a UFO (and then the UFO's lights resemble a truck's!)
The unnerving way the aliens bring back objects like the planes flown in Flight 19... but not the people until the end. And leaving them in places where those planes and ships do not belong.