During Shadow's story a GUN Agent radios in to telling you to head to Wave Ocean. You hear the actor mess up his line and repeat it again.
Rouge gets radioed in by a GUN agent, the agent asks if Rouge retrieved the Scepter of Darkness but the subtitle says Book of Darkness.
At one point during Eggman's conversation with Elise you can see his lips continue moving after he finishes his sentence.
Silver the Hedgehog giving Princess Elise the blue Chaos Emerald in the past and creating a time loop causes a major plot hole to the entire franchise because how could there be a blue Chaos Emerald in any of the previous Sonic the Hedgehog games when Princess Elise has had it in her possession for a decade.
In the opening to Shadow's story, after he communicates to the G.U.N. commander on the radio, he teleports, which wouldn't be a problem except for the fact that he doesn't have a Chaos Emerald and as it's been established in previous games, Shadow can only teleport when he has a Chaos Emerald to initiate Chaos Control.
Blaze the cat was previously established in "Sonic Rush" on the Nintendo DS to be from a different dimension but in this game, that was retconed to her being from the future. Due to this, Takashi Iizuka had to make a statement saying to ignore this game as she is officially canonically from a different dimension.
Shadow now works for G.U.N despite them killing his best friend 50 years ago and the commander trying to kill him in the last game. It's never explained why he comes to this absurd decision or what has changed his mind on the organization.