Do I need to register with the Police upon arrival in Serbia?
The answer is no if you are coming on a short stay, e.g. vacation.
If you intend to apply for residence, you should register with the Police and you will be given a so-called 'white card' that you can then use for official residence purposes.
All of this applies unless you have a powerful enemy in Serbia in which case you will be arrested and deported for spitting on the sidewalk; so registration with the Police will be the least of your problems.
Western embassy advice on this subject is simply wrong.
UPDATE 23 May 2022
Can the Police search me at random?
It is debatable whether they have lawful authority to do so; but as a matter of practice they consider themselves to have unlimited authority to search anyone they want at any time.
It is not unheard of for the Police to raid a nightclub and force everyone to lie face down on the floor with their hands on top of their heads; and then to systematically search everyone for drugs, shouting and kicking anyone who wriggles around during this period of enforced lying face down on a hard floor. Obviously this is very unpleasant.
These things should not be happening; but modern Serbia is a country with a number of flaws and occasional unreasonable aggression on the part of the Police is one of them.
If this happens to you as a visitor, seek to hold your passport on top of your head while you are lying there, to minimise the risk of casual Police violence.
Quite aside from questions of human rights compliant European standards of policing, this approach is poor drugs policy, because rough Police treatment of civilians alienates the relationship between the two, and hence makes civilians less likely to cooperate with the Police in order to identify and capture the criminals who run the most dangerous drug gangs. The Police cannot undertake this important function if the people of Serbia consider the Police as enemies to be avoided by reason of their arbitrary aggression.
Let us hope that Police practice becomes more moderate and wise in working to manage Belgrade's principal social problem, namely the pandemic of recreational narcotics some of which are very harmful but many of which are not.
Must I carry my ID in Serbia?
Yes. You must carry your passport with you at all times. If you cannot present it to the Police when asked, then the Police are likely to arrest you. You will be taken to a Police station and fined (typically a sum such as 100EUR) after an extended period of waiting during which you will be required to prove who you are (for example, by showing a copy of your passport on your mobile telephone). As is typical in Serbian police stations, you will not be allowed to leave the Police station until the fine has been paid in cash. You may be frog-marched to the cash point.
What are my rights in a Serbian Police station?
In principle, you have the rights -
to the presence of an interpreter
to the attendance of a lawyer paid for by the state
that your consular officials be informed
In practice you are unlikely to be afforded any of these rights. Instead you will be given papers to sign in Serbian Cyrillic and instructed to pay a fine. It is usually wise simply to sign the documents and pay the fine, as long as you can understand how much the fine is. That is how you will achieve release from custody most quickly.
--- The PALADINS