Chenobyl I thought they were running tests on the reactor with most of the safety systems switched off, and suffered a runaway reaction made worse by a side effect of the emergency shutdown procedure. Which resulted in an explosion that blew the top off the reactor. Not sure what benefit modern reactor protocols would have compared to the older ones they were using at Chenobyl. Are they now written in some sort of special ink so that they work even if you ignore them, and bugger about with an unstable reactor with the safety systems switched off?
You could also mention Three Mile Island, Windscale, and Fukushima on the down side of the nuclear industry, although I believe burning coal instead of using nuclear power to generate electricity. Actually releases more radioactivity in the long run, the leakage just tends to be less exciting.
Right, I'll bite. Reactor 4 was being run down to test something or other, on the morning of the test a full test team was present however there was a failure of a coal power station near Kiev, as such the grid controllers requested Reactor 4 was left at full power to cover the gap, it was. The day shift and test crew went home, the night team came on expecting to just babysit a reactor effectively at idle, instead for various reasons they performed the tests unprepared.
In order to do this all of control rods, including rods normally locked in place were removed, the reactor started to run out of control and was in a totally unsafe configuration and began to run out of control, as such the operators/computer (unknown which) began a SCRAM, an emergency shut down. Due to a design flaw of the RBMK reactors used in the USSR at the time as control rods entered the reactor they actually sped up the reaction for a short while before they would begin to slow it down. Due to another flaw in the design at the high temperatures steam voids could form, they did. These voids and high pressure steam caused the fuel and control rods to buckle which had the effect of leaving the rods actually raising the output and caused a massive power spike, two steam explosions within the reactor followed. Unlike modern and Western reactors the RMBK design did not contain the core within a pressure vessel, it just exists in a giant cylinder with a 2000t cap on it held on with not much more than gravity and bolts. The steam explosions threw that cap off and with it much of the reactor core.
Fast forward to today. Modern reactors all exist within a sealed pressure vessel, which is why radiation release from Three Mile Island and Fukushima were so much less than Chernobyl and why Windscale released so much more than the severity of the incident, non contained reactos are not a thing anymore. It is no longer possible to just withdraw all the control rods you wish to run a test to please your superiors. The RBMK design is so dangerous that as part of Lithuania's membership of the EU it was to close its remaining RBMK reactors and the EU would help with replacing them.