I watched so many cartoons about Sylvester & Tweety but I cannot remember in any of those seeing Sylvester catching Tweety; does anyone remember in any serious cartoon about them somehow Sylvester catching Tweety?
Sylvester actually grabs Tweety several times. Here's one example from Tweet and Lovely (1959) (it's around the halfway mark):
And here's another (I Taw a Putty Tat from 1948) where he not only grabs but eats him but Tweety manages to escape from his mouth. Sylvester grabs him and almost eats him yet again later, but is foiled again when Tweety tricks him into eating alum. And he grabs him and almost eats him yet again in Birds Anonymous from 1957 (found here, dubbed in Italian), where alum is also employed.
As for Sylvester actually eating and killing him, that also happened but it turned out to be a dream. From the synopsis of the final episode of The Sylvester & Tweety Mysteries from 2000:
Sylvester finally eats Tweety, but he's thrown in jail and the show is canceled resulting in the world crashing down on him. This turns out to be a dream much to the relief of Sylvester.
There is another cartoon called "Greedy for Tweety" (which was basically the follow-up to "Birds Anonymous") and while Sylvester, Tweety and Hector were in the hospital at that point, Sylvester did however manage to succeed in eating Tweety at one point.
But once Granny found out that he ate him, she did an X-ray on Sylvester and took him to the Puddy surgery (Emergency Room), where she proceed to cut him open to get Tweety out (from there on he came out with a large bandage around his stomach).