Got a cut on my hand at work (I work as a waiter). Went to see GP, medical bill was $120 for stitches and some meds. Should I claim under company insurance or file work injury claim?

Since this is a work-related injury, your employer will need to file a work injury incident report with MOM. MOM will conduct an investigation, and then issue a Notice of Assessment. Once MOM declares you've suffered a work-related injury, your company must compensate you. Under Singapore law (the Employment Act, specifically), you are legally entitled to receive work injury compensation. Note that this is an entitlement (meaning you cannot be denied compensation), and not merely a benefit.

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Your employer will legally have to compensate you $120 for your medical bills. In addition, if you received medical leave for this cut, they will have to pay your wages while you're on MC. It doesn't sound like you suffered a disability, but if you did, make sure to sound that out. The law provides for mandatory compensation for work-related disabilities.

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If your employer has a Workmen Compensation policy, the policy will cover these expenses. If they don't, then they can claim under their Company Health Insurance. Anyway, these are back-end processes which shouldn't concern you too much. You can focus on notifying your HR/boss and then rest to get better; your company should do the rest since you're injured. Have a good recovery and feel better soon.