Aspirin, acetaminophen, ibuprofen, and naproxen are pain medications available over the counter. There are also dozens of other non-steroidal anti-inflammatory medications, and pain meds for neuropathy that are not narcotic. If you have particular drugs in mind, just let me know.
Darvocet is propoxyphene napsylate plus acetaminophen. Tylenol 3 is codeine plus acetaminophen. Both of these show up on drug tests if that is your concern. Codeine shows up on all tests as an opiate (the NIDA-5). Propoxyphene is only on the expanded tests. The NIDA-5 tests for cocaine, cannabinoids (marijuana, hashish), amphetamines, phencyclidine (PCP) and opiates (codeine, heroin, morphine). The expanded tests also include barbiturates (phenpbarbitol, butalbital), hydrocodone (Vicodin, Lortab), Benzodoazepines (Xanas, Valium), Methaqualone (Quaaludes), Methadone, ethanol (alcohol), MDMA (Ecstasy), and propoxyphene (Darvon, Darvocet).