1. Start a DVD club membership (a-la-Blockbuster)
2. Rent an appartment
3. Register a bicycle
4. Rent an automobile
5. Purchase a long-term railway pass
6. Register your child at a public (or private) school
7. Apply for social insurance.
If you are riding two to a bicycle and the cops stop you for that (which they may), they may ask to see your Foreigner ID.
They may ask for your Foreigner ID at any time, in fact...
In Japan, the cops are always polite, but they would NEVER hesitate to ask for your Foreigner ID card...