(at around 9 mins) When the governor enters the Smiths' home (with the band playing) we see, from the inside, Ma closing the door almost shut. When the scene shifts to outside the house, Ma is again closing the same door.
(About 45 mins) Smith and Susan Paine have a conversation wherein Smith's hat, held in his hand, is the camera's sole focus for about 30 seconds - at the expense of the characters conversing. And yet when Smith leaves the room, he is without hat, and the hat is nowhere to be seen.
(at around 30 mins) Twice, Saunders says that Jefferson Smith is going to go "up" to Mt. Vernon, the home of George Washington. Mt. Vernon is approximately 15 miles south of the Capitol in Washington [you can check that using Google maps], and is right on the Potomac River downstream from Washington, so it is not "up" in a north-south sense nor in the sense of elevation. The script should have had Saunders saying that Smith was going "down" to Mt. Vernon, which is how anyone living or working in Washington would have put it.
The President of the Senate repeatedly recognizes Jeff Smith on the floor of the Senate as "Mr. Smith." Senators are not recognized by name but as junior or senior senator from the state they represent.
Under the Standing Rules of the Senate governing debate, Senator Paine would not technically have been allowed to attack Senator Smith's character and accuse him of graft. The rule states: "No Senator in debate shall, directly or indirectly, by any form of words impute to another Senator or to other Senators any conduct or motive unworthy or unbecoming a Senator."
(at around 21 mins) In the montage as Jeff Smith wanders off after arriving in Washington, we see the display of the Declaration of Independence at the National Archives, and then supposedly a close-up of the document - only the "S"'s are written in modern style script and don't look like "F"'s as in the original document.
(at around 17 mins) At the train station, Jeff Smith is approached by Susan Payne and three other women. They ask for a dollar contribution each for the Milk Fund. Jeff Smith says "five dollars". It should only be four dollars. Soon after, reporters ask him about the women in Washington. He responds that four came up to him at the train depot.
(at around 18 mins) When Smith arrives in Washington on the train, he's seen walking towards the exit with a porter behind him carrying his bags. The next shot shows the same porter coming into the station carrying someone else's bags.