Are wage expense considered a current liability?

Generally NO, wages are an expense. The only exception to the rule is if a company has "wages payable" which is wages that they owe but have not yet paid, "wages payable" is a liability until they are paid. Once paid, the account is closed into wage expense and is listed under the asset column of the Trial Balance sheet, until the end of the accounting cycle when expense accounts are closed out for the year end.