I'll never forget the day *(on) which/that* I met you.
1.
a. I'll never forget the day on which I met you.
b. I'll never forget the day which I met you on.
→I know that sentence a is correct, but I wonder whether sentence b is correct or not? I know that some people think that it's bad to end a sentence with a preposition, but is it true? Do you native really care if you end a sentence with a preposition or not? When I end a sentence with a preposition, maybe it is not good, but does it mean that the sentence is wrong, like sentence b?

2.
a. I'll never forget the day that I met you.
b. I'll never forget the day on that I met you.
→Are the two sentences both correct? Are there any differences between them? Which is more common?

Thanks!Furthermore,
Can I say it like this? "I'll never forget the day that I met you on."

Question

I'll never forget the day *(on) which/that* I met you.
1.
a. I'll never forget the day on which I met you.
b. I'll never forget the day which I met you on.
→I know that sentence a is correct, but I wonder whether sentence b is correct or not? I know that some people think that it's bad to end a sentence with a preposition, but is it true? Do you native really care if you end a sentence with a preposition or not? When I end a sentence with a preposition, maybe it is not good, but does it mean that the sentence is wrong, like sentence b?

2.
a. I'll never forget the day that  I met you.
b. I'll never forget the day on that I met you.
→Are the two sentences both correct? Are there any differences between them? Which is more common?

Thanks!Furthermore,
Can I say it like this? "I'll never forget the day that I met you on."

Answer

Rewrite the sentence in smaller parts.  
I met you on a day.  I'll never forget that day.
I'll never forget the day that I met you on.

We would not write 'I met you a day'
So 'I'll never forget the day that I met you' isn't accurate.

We don't always use correct grammar when speaking.  I would say 'I'll never forget the day I met you.", but I would not write it that way.

Answer

You may end a sentence with a preposition. Native speakers do it all the time naturally. I would say:

I'll never forget the day that I met you.
I'll never forget the day I met you on.

In fact, the "that" isn't even necessary. You can say: I'll never forget the day I met you.

Answer

Hi Susanne -

Thanks for your response. And you said, "I third Michelle's response." In fact, I know you are trying to express. But, only "second" can be used as a verb, which means to agree, but "third" cannot be used as a verb, according to many dictionaries. Is such a usage of "third" common in English? Thanks!

Answer

I was taught not to end a sentence with a preposition but I now know that is a false rule. Sometimes it sounds fluent and other times, it does not. You just have to try it out each time.

And I third Michelle's response.

Answer

I second Michele's answer.
As far as prepositions. Yes you may end a sentence with one. But it has to sound right. And that is why 1b would sound odd.