In my opinion...
In answer to the OP, 99.9% of the time you don't.
The only time I can think that you might is if you're using it as some kind of negotiating tool, however this is a high risk approach with a good chance of backfiring. Maybe if you're close with a non management colleague you might mention it but it could still be risky.
I'm with most of the others regards references. Most potential employers wont expect to get detailed references, if any, from a current employer. And as others have said many companies will now only confirm employment dates not much more.
References from previous employers, college professors or former colleagues are a better idea.
That said I’ve broken most of these rules but under specific circumstances.
I once left a job application (or may have been an add) on my desk after we’d had some layoffs. It wasn’t particularly intentional but wasn’t just carelessness. My Technical Director whom I had a pretty good relationship with saw it and assured me there wouldn’t be layoffs in our department, plus I think it may have something to do with my above inflation pay rise & an unheard of bonus a couple months later. However, I wouldn’t dream of doing this at my current employer.
Also when I did eventually leave that company if was to emigrate to be with my wife, not for another job, so I was fairly open with them and although my formal resignation was short and sweet I’d verbally let them know what was happening some time before, everyone except the financial director was fine with it. I got my boss and a more senior colleague to give me written references, also several people I’d worked with from different companies in the US agreed to be references. Again though it was a somewhat unusual situation and even if in the same circumstances at my current employer I wouldn’t say anything.
thread731-180043 & thread731-178435 though talking about resignation letters have some relevance.