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I think it would take extreme circumstances for a stainless razor to rust but if I owned one I'd likely still give it a coat of oil. Just paranoid I guess :)

Stainless sinks will likely have more chromium etc to prevent rust but leave a wet tin can on a stainless sink overnight & tell us if the sink rusts :)

I just wipe the blade with tissue paper pre...

Most straights are carbon steel & even stainless can rust tho not as easily.

I oil razors after every shave & razors in storage once a month. High humidity here.