My mother is sick to me: the possessive Serbian builds without a possessive
A Russian speaker reaching for possession reaches for у меня plus a genitive noun. Serbian's everyday equivalent skips both: the possessor turns into a bare dative clitic - ми, ти and му - clipped straight onto whatever is possessed, with no preposition anywhere in sight. Missing this one habit produces sentences that stay grammatical, get understood, and sound just slightly foreign every single time.
Take an ordinary Serbian sentence that says a mother is sick, and something is missing that an English or Russian speaker expects to find: there is no word anywhere in it that means my. The possessor is there - it has to be, the sentence would be meaningless without it - it is just not built out of a possessive at all. It is built out of a dative pronoun, clipped onto the sentence the same way a reflexive or an auxiliary would be, and a Russian speaker's instinct almost never reaches for that option first.
The dative that isn't a recipient
Grammars call this the free dative or the possessive dative: a dative pronoun attached to a clause not because anyone is receiving anything, but to mark whose the following noun is. It shows up most naturally with the things a speaker feels are closest to them - family members, body parts, health, everyday belongings going right or wrong - and it sits directly next to the noun it possesses rather than in front of it as a separate word would.
Мама ми је болесна.
My mother is sick.
Ми is the entire possessive - there is no separate word anywhere in the sentence for my.
Where Russian reaches instead
Russian's closest everyday equivalent for exactly this kind of sentence is у меня plus a genitive noun, or simply a possessive adjective like моя sitting in front of the noun the ordinary way. Neither option involves a dative pronoun glued onto the sentence, so the instinct a Russian speaker brings to Serbian is not wrong, exactly - it is just a different, equally grammatical tool being reached for out of habit.
The calque to watch for is reaching for код мене - literally at my place - to do the job у меня does in Russian. A sentence built that way is grammatical and gets understood, but it reads as though the mother happens to be sick at the speaker's house, not as a plain statement of whose mother she is. A bare dative clitic, not a prepositional phrase, is what a native sentence reaches for here.
Not a replacement for every possessive
This is not a claim that Serbian abandoned possessive adjectives altogether - мој отац is perfectly ordinary Serbian, and nobody would blink at it. The free dative is an additional, often preferred option specifically in this narrow territory: family, body, health, and things breaking or going wrong. Reach for it there and a sentence sounds native; reach for a possessive adjective instead and nothing is actually wrong - it is simply the more distant, more formal-sounding of two correct choices.
Ауто ми се покварио.
My car broke down.
Ми (dative) comes before се (reflexive) here, exactly the order the clitic-queue article on this blog already fixes - the two clitics are doing two different jobs and still have to stand in that sequence.
None of this is exotic grammar reserved for advanced learners - it is how a beginner hears about a friend's sick relative or a broken-down car within the first few weeks of actually listening to Serbian. The habit worth building early is small: before reaching for код мене or a possessive adjective out of pure reflex, check whether a bare dative pronoun would already say it, more naturally, on its own.