Bedbugs: Eww! I think I'd puke if I ever saw one!
Mattresses from second-hand and thrift stores: I'm surprised that's even legal!
And from the street! EEK! You'd think it's obvious that a mattress that sat out overnight would immediately get moldy from the dew. How people can sleep on something they just got from the street is beyond me!
The cause of that has got to be city policies that don't allow immediate disposal of mattresses in the weekly household trash collection. I've seen enough mattresses on the street that I think I'm going to write to my city council member about this. The sanitation department should be empowered to pick those things up immediately wherever they are seen, regardless of anything else. God!, talk about a huge health hazard!
Especially since people can pick up bugs visiting others who have them. Yow! Lucky me having a limited social life, oh the glorious diseases and disease vectors I never get to see....!

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Also, mattresses that sit around on the street become a great nesting-place for mice, rats, and God knows what else, all of which carry fleas, which in turn carry bubonic plague. Yeah, definitely a need to get cities to change their trash collection practices to deal with mattresses, or sooner or later we are going to regret it bigtime.
Re. bleach & dioxin: I've heard that chlorine breaks down into chloramines, which are somewhat toxic but hardly as bad as dioxin, and I have not heard anything about chlorine breaking down into dioxin. People have been using bleach for the better part of a century without ill effects if used correctly, so I wouldn't worry about it. The bugs the bleach would have killed can bite you harder than the bleach.
Probably the UK/Ireland procedure of oxygen bleaches and hot washes is more than sufficient in most cases, with a occasional dose of chlorine bleach once in a while if you find it's necessary.