Select the INCORRECTLY spelt word.

The correct option is Bycycle.

Let's analyze the spelling of each word provided in the options:
1. Bycycle: This is spelled incorrectly. The correct spelling is Bicycle, which is a vehicle with two wheels propelled by pedals.
2. Academy: This is spelled correctly. It refers to a place of study or training in a special field.
3. Mystery: This is spelled correctly. It refers to something that is difficult or impossible to understand or explain.
4. Justice: This is spelled correctly. It refers to just behavior or treatment.

Therefore, "Bycycle" is the incorrectly spelt word.