Samira Shackle in New Hummanist:
Penicillin
Perhaps the most famous accidental discovery of all is penicillin, a group of antibiotics used to combat bacterial infections. In 1928, Scottish biologist Alexander Fleming took a break from his lab work investigating staphylococci and went on holiday. When he returned, he found that one Petri dish had been left open, and a blue-green mould had formed. This fungus had killed off all surrounding bacteria in the culture. The mould contained a powerful antibiotic, penicillin, that could kill harmful bacteria without having a toxic effect on the human body. At the time, Fleming’s findings didn’t garner much scientific attention. In fact, it took another decade before this drug was available for use in humans. Retrospectively, Fleming’s chance discovery has been credited as the moment when modern medicine was born.
Pulsars
In 1967, astronomy graduate student Jocelyn Bell noticed a strange “bit of scruff” coming from her radio telescope. It was a regular signal coming from the same patch of sky, of a type that no known natural sources would produce. Bell and her supervisor, Anthony Hewish, ruled out sources of human interference – other researchers, television signals, satellites. None explained the signal, and the scientists wondered if they had detected a sign from aliens. This was ruled out when another was located in a different part of the sky: it seemed unlikely that two sets of aliens would simultaneously be trying to communicate with Earth. In fact, it was the first discovery of a pulsar (pulsating radio star), a highly magnetised, rotating neutron star that emits a beam of electromagnetic radiation. Pulsars, which had been predicted three decades earlier but had never been actually observed, indirectly confirm the existence of gravitational radiation.
More here.