Continuity mistake: Before Marshall proposes the gold foil is still on the bottle of champagne. He sets it down with the foil still on and they have sex on the kitchen floor. When they sit up, he grabs the bottle of champagne again and the foil is already off. (00:04:50)
Other mistake: At the beginning of the series, Ted starts telling the story of how he met the mother of their children. When he starts describing how he met Robin, he doesn't reveal that it was not their mother until the end of the pilot. But children should have realised it wasn't their mother when Robin said she was a journalist and worked as a TV presenter. The kids knew their mom wasn't a journalist, so they shouldn't have been surprised at the end of the episode when Ted said "this is how I met your aunt Robin".
Character mistake: When they see Robin on the news at the end she says "Metro 1 News" instead of "Metro News 1". (00:12:20)
Continuity mistake: In the pilot episode we see the Son and Daughter listening to the start of the story on how Ted met their mother. In the final episode of the show it's revealed that the whole story has taken place in one sitting. However, the clothes the kids are wearing and the items around them are different in the pilot episode to every other episode. The sofa is a different design, the cushions appear etc. (00:00:45)