"Zijn" in "Ik vind dit zijn beste schilderij."
(This is the sentence I saw somewhere else.) I thought "zijn" is used for plurals, but in this sentence all nouns and pronouns are singular, why, then, the verb is in plural? Can I say "Ik vind dit is het beste schilderij"?
Exactly, so the meaning is: "I think this is his best painting", or: "I find this his best painting".
And to add to that, you cannot say "Ik vind dit is het beste schilderij" but this works: Ik vind dat dit het beste schilderij is = I think (that) this is the best painting.
I am not a native Dutch speaker, but I would be glad to explain some Dutch grammar here.
No, you cannot leave 'dat' out in Dutch in the same way that 'that' can be (and often is) left out in English.
When 'dat' marks the beginning of a dependent clause, the word order changes and without this "dependent clause marker" the context is lost.
Dit is het beste schilderij = This is the best painting. (independent clause)
This can also be translated as: This is his (or her) best painting — without using "zijn" (his) or "haar" (her) — if the context is already previously stated or known.
Or if further context is given:
Dit is het beste schilderij van Van Gogh = This is Van Gogh's best painting.
Ik vind dat... = I think that... (incomplete thought)
...dat dit het beste schilderij is = ...that this is his best painting (dependent clause)
Note that the verb (or verbs) move to the end of a dependent clause in Dutch:
Ik vind dat dit het beste schilderij is = I think this is his best painting.
Other ways to express this:
volgens mij: in my opinion
Volgens mij is dit het beste schilderij = I think this is the best painting.
Note that subject-verb "dit is" is inverted to verb-subject "is dit." Inversion happens whenever the independent clause is preceded by a word, phrase or clause other than a conjunction. In this case, the prepositional phrase "volgens mij" (literally, "according to me").
ik denk: I think
Dit is denk ik het beste schilderij = I think this is the best painting.
Here, "ik denk" appears in the middle of the sentence as if it were a parenthetical. And note that it is inverted:
Dit is (denk ik) het beste schilderij.
Normally "denk ik" is used in more elaborate constructions, however, such as:
Dit is denk ik het beste schilderij dat ik ooit heb gezien = I think this is the best painting I've ever seen.
And of course the sentence we began with:
Ik vind dit zijn beste schilderij = I find this his best painting, i.e. I think this is his best painting.
Hope this helps!
p.s. I'm sure the Dutch community will correct me if I have made any mistakes here.