yup, debit card can too just make sure your debit card register for secure online shopping.

How to Buy Using your debit cards
1. login to your live account, Edit your Contact info then asign your country region to Singapore and Save.
2. login to your www.livexbox.com, Go to MyXbox > Manage Payment Options > Add New Credit Cards
Enter your Cards Holder Name
Address - Local Address where you apply for your debit card account, only change your postal code to singapore ex. 640936
Add your Debit Card Number and 3Digit Security Code.
and Hit Save.

3. Go to your marketplace buy apps.

each device are tied with your own live ID so applications cannot share, you have to purchase with your own live id.