Which is correct ......

[butterfly76]

I should like to apply for or I would like to apply for?? thanks

[mrs_overall]

Would

[benny3008]

'I would' I would say personally. :-/

[dotty.]

I am submitting my application for:

[dotty.]

using 'like' sounds unprofessional

[dotty.]

put:
I am submitting my applcation for the role of whatever for your consideration

[factor-fiction]

If you are not sure (and would and should is not an easy one) I would (should?) change it to something else.
I suggest:
I wish to apply for ..or
I am applying for...

[Coldicote]

I agree with these replies. One further point, the word 'should' implies 'ought to'.

[dotty.]

and 'would like to apply' suggests you might, but there's a reason you can't lol

[coccinelle]

Years ago we used to use should instead of would for the first person (I should and We would) just like we used I shall and We shall instead of I will and We will. Due to using I'd and I'll etc we now use would and will for all persons. Should now means ought to and Shall in questions as in Shall I open the window?
As factor suggests I'd use I wish to....