How to hold a tap in the tailstock?

If you make a die (or tap) holder don't make the slide rod for an MT taper. Invariabley you will buy another lathe with a different size tail stock. Just use a piece of solid rod as big as will fit in your drill chuck. If you already drilled and bored your die holder too big for that you can always turn down one end of a properly sized guide rod to fit your drill chuck.
 
I guess i do it differently from everyone else.
I put the tap in a drill chuck mounted in the tail stock and get it super tight.
Put the lathe in back gear at dead slow speed.
Loosen the tail stock brake.
Turn on the lathe and push the tail stock to engage the tap into the work.
The tap and tail stock will then advance by virtue of the threads automatically advancing into the work.
Stop the lathe in time and reverse it. The tap and tail stock will back out on it's own.

This has worked flawlessly many times for me and I have never broken a tap.

Works on my machine.
 
I was originally trying to tap under power, I am learning from this thread that is not advisable. Best to do it by hand it sounds like.
Tapping under power is done every day. I do it on my Sheldon and on my Seneca Falls machines, and have yet to break a tap. I insert the tap in a Jacobs Super Ball Bearing chuck, and put the lathe in the lowest speed on the back gear (about 40 rpm's) Slide the tailstock to engage the tap, and tap to depth.
Every once in a while a tap will spin in the chuck. If it does I loosen the chuck and back it away. Then I put a tap handle on the tap and attempt to turn it by hand. If it goes easily I remove the handle and bring the chuck forward. This time I tighten it just a bit more. 99% of the time all goes well.

On the other hand if the tap does not cut easily and smoothly with the tap handle I remove it from the part and inspect it. Sometimes the tap is just old and dull. Replacing it with a new one usually cures the problem. If a new tap has a hard time it usually means you've hit a hard spot. Then it takes a bit more finesse and finishing it by hand becomes the best option. Advancing about half a turn then backing off (just like when using a hand tap) has always cured the problem. It might be a bit slower, but breaking a tap would be a lot worse.
 
I guess i do it differently from everyone else.
I put the tap in a drill chuck mounted in the tail stock and get it super tight.
Put the lathe in back gear at dead slow speed.
Loosen the tail stock brake.
Turn on the lathe and push the tail stock to engage the tap into the work.
The tap and tail stock will then advance by virtue of the threads automatically advancing into the work.
Stop the lathe in time and reverse it. The tap and tail stock will back out on it's own.

This has worked flawlessly many times for me and I have never broken a tap.

Works on my machine.

Same here. Only thing I do different is not chuck the tap super tight. I intentionally go a bit lighter than normal so if things go bad the tap starts to spin but not break. The tap body being hardened and smooth it doesn't tear up the jaws. Only use relatively new and sharp taps and hole drilled for 60% thread. I keep an close eye on it going in and at first sign of slippage will reverse, clean chips and repeat. Still I don't use my nice Albrecht keyless instead I have an old Jacobs chuck for tailstock tapping. Another key is having a 3 phase geared head lathe you can instantly reverse (plug)
 
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