Only #1 is grammatically incorrect, although the question itself is grammatically wrong. It should read "Which of the following sentences is grammatically right." There is more than one sentence so it needs to be "sentences", not "sentence".
Number 1 is wrong: it should be "This will remain between you and me" The object of the preposition "between" needs an accusitive form.
Number 3 is also wrong, but it is not grammatically wrong. "Lays" is a transitive verb; you need the intransitive "lies" so the sentence should say "He lies on the couch" That is an error in vocabulary, not grammar.
As well, the "i" in number 2 and the "he" in number 3 should be capitalized. Capitalization is not grammar.
Not technically, but it means something different than you might think. "She wanted him to remain silently." means that she wanted him to stay where he was while being silent. "She wanted him to remain silent." means that she wanted him to continue to be silent, regardless of location.
These gallows remain unused.
No you do not capitalize artist in a sentence
Fanfare, used in a sentence: The dramatic fanfare at the glorious crowning of the Queen will remain in my memory forever.
A person and a shirt
It is not a sentence. There is no main verb.
Not technically, but it means something different than you might think. "She wanted him to remain silently." means that she wanted him to stay where he was while being silent. "She wanted him to remain silent." means that she wanted him to continue to be silent, regardless of location.
A sentence could be:"i was remain to believe" does not make sense and is bad grammar.
No it is not a sentence. You might try: I will cherish you forever.
We should remain silent when others are shouting.
These gallows remain unused.
i will reman in this small town forever!
I remain undaunted at the task.
The Mayan hieroglyphics remain undecipherable.
Further Leave to Remain Other
No you do not capitalize artist in a sentence
The law is dogmatic and will hopefully remain unapproved.