If my wife fell in a supermarket and injured her wrist and hand, how would she go about collecting for medical and time off work pay?

You should consult a personal injury attorney immediately. Typically a store would have insurance to cover such an incident. If the store refuses to admit fault, that is where you need an attorney to help you determine fault and how to go after the store for repayment.

Your wife's personal injury claim should be filed with the supermarket's insurance carrier.  When your wife completes her medical treatment and is released by the doctor or is declared to be permanent and stationary which means having reached a point in her medical treatment where no further improvement is anticipated, she should obtain her medical bills, medical reports and documentation of any wage loss.  The personal injury claim filed with the supermarket's insurance carrier should include these items.  Compensation for the medical bills is straight reimbursement.  Compensation for wage loss is straight reimbursement.  The medical reports will document the nature and extent of your wife's injuries and will be used to determine compensation for pain and suffering which is an amount in addition to the medical bills.  If your wife is dissatisfied with settlement offers from the supermarket's insurance carrier, she should reject the settlement offers and file a lawsuit for negligence against the store.  If the case is settled with the insurance carrier, NO lawsuit is filed.  If the case is NOT settled with the store's insurance carrier, your wife will need to file her lawsuit for negligence against the store prior to the expiration of the applicable statute of limitatiions or she will lose her rights forever in the matter.  The statute of limitations varies from state to state, but since the accident just happened a few weeks ago, there is plenty of time remaining as the statute of limitations is probably one or two years.  Without knowing the state in which you live and/or where the store is located, it is not possible to give you the  exact length of time for the applicable statute of limitations.