I was wondering if I see attractive lady close to my age in grocery shop, I will approach her and say something (not personal) about what is going on and will watch body language. if I see signs she doesn't feel like talking, I will apologize and move away.
BUT, what if she is smiling than I can make a normal conversation. I would like to ask for SMS only to continue conversation but in most cases I noticed that women were defensive whi h is understable. Not to ask anything, there is possibility that I would never see her again.

I think the 'grocery shopping' approach has been over-hyped.
Gals don't 'fix up' to go marketing.
Which may be why they seem defensive.

If tis where you always shop, why not just a few smiles
and let them be the aggressor, if they wish ;-)

Briefly talking in a supermarket and asking for a woman's phone number after that???
I often don't even give my phone number when I'm going on a date with a guy.
To me it screams "Creep!!!" if a mere stranger whom I happened to have talked to in the supermarket asks for my phone number.
Now if you'd see someone more often and have conversations... then you could suggest having a cup of coffee. If at all possible in a coffee corner in / next to said supermarket. That lowers 'creep' levels and feels more natural.

you are a 'very uptigte' sort.

Some Men give out their phone number like they are giving out lollies .. so why not give her your number .. if she is interested she may contact you .... life is about taking chances .. good on you for making an effort

mtt, I guess you can go for it but do it in a sensible manner.