I would imagine that following the Norman invasion, it might be possible that Anglo-Saxon nobles or warriors might make attempts to take back the English throne. Were there ever any cases of uprisings or resistance?
After the Battle of Hastings, William has to wage a series of campaigns to actually secure and consolidate his hold on the English throne. To some extent he's fortunate that a significant number of the English nobility are killed at Hastings, and the rest are quickly replaced by Normans. The rightful heir to the English throne, Edgar Ætheling, surrendered to William at Berkhamstead some 2 months after the battle. This doesn't go entirely smoothly. For a while, a resistance campaign is run by Hereward the Wake, who according to the Gesta Herewardi, returned from exile in 1069 or 1070 to lead a campaign against the Normans alongside the Anglo-Saxon Earl Morcar of Northumbria. They operated out of the Isle of Ely in the swamp area of the Fens which confounded Norman cavalry tactics. Morcar was eventually captured, but the fate of Hereward is still unknown.
In 1068, the city of Exeter revolted under the leadership of Gytha Thorkelsdóttir, the mother of Harold Godwinson. William laid siege to the city, which eventually surrendered on the condition that there be no reprisals, and its rights and privileges under Anglo-Saxon law still be upheld. In return, William built Rougemont Castle in the city to maintain control. Other rebellions occurred at Dover, Hereford, Nottingham, Durham, York and Peterborough, and castles were eventually built at all. Castles were central to the Norman control of England. They allowed a relatively small military presence to control a large area, and a symbol of Norman power to the local populations.
Castles didn't always work though. Perhaps the most famous resistance to the Normans resulted in the 'Harrying of the North.' The former 'Danelaw' areas of the north had been granted relative political autonomy by King Edgar in the 970s in return for their loyalty. Both ethnically and culturally, they were distinctly Anglo-Scandinavian, and actively resisted the Normans who sought to abolish their nominative independence and incorporate them fully into England. William's appointed earl was murdered by the son of his Anglo-Saxon predecessor, who in turn was killed by another earl who then pledged allegiance to the deposed Edgar Ætheling. When William made Robert de Comines earl of Northumbria, he and his entourage were ambushed by an angry mob in Durham and massacred. In response, further rebellions break out in the West Saxon heartlands of Dorset, and a Danish fleet harries the east coast before seizing York and declaring for Edgar.
In response, William marches to the north with an army, and basically kills and burns everything in his path, cutting a massive swathe of destruction which is still clearly visible in Domesday Book in 1086. In what has been sometimes called a genocide, over 100,000 people died either by the hand of William's army, but moreso through famine as the army destroyed farms, burned stores and seized food. Even in 1086, Domesday records that 60% of the Yorkshire North Riding was desolate wasteland, 66% of manors had been destroyed, and only 25% of the 1066 population remained.