S1E1. In the final scenes of the episode, one of the characters falls, and you can clearly see the rope they were holding on to, falling with them. In the next scene the rope is shown hanging from the railing.
When Kentaro first runs up to his daughter Cate at the camp in San Francisco; there is an ambulance at Cate's right-hand side. Then in the next cut, there is now a Red Cross tent on Cate's right-hand side. Then in the next few cuts it goes back and forth between the ambulance and the tent.
The WTA Airliner touching down is a Boeing 787 Dreamliner, the very next view shows only the wheels, which are now 4 main landing gears of a 747.
Due to space constraints and small average apartment sizes in urban Japan, most apartment front doors open OUTWARDS to the hallway, not inwards into the apartment.
The SFPD cars featured when Godzilla attacks on the Golden Gate Bridge have the completely wrong logo. Actual SFPD car logo is the 7-pointed SFPD star, reading "SFPD".
The taxi Cate is in shows the price of 730 yen, which in 2015 is incorrect.
The respirators being worn by the scientists in 1959 Kazakhstan aren't from that period. These are clearly US M40 or M45 1980's or 1990's era respirators being worn with a filter on both sides. These respirators are normally worn with the filter on the right or left depending on if the wearer is left or right handed. Dual filter respirators didn't enter service until the 2000's. They should be wearing a lighter grey coloured M9 respirator which was phased out in 1959.
In 1959 Kazakhstan they drive a beaten up Series 2 Landover, which wasn't launched until 1958. Finding one in Kazakhstan in that state a year after release is highly doubtful.
A damaged Salesforce tower is shown in a fly over of the city just before a title card stating the year is 2014. The tower's foundation was just being poured in 2015.