It's a matter of personal taste, to some degree, and will mainly depend on your target audience, the style guide you're using, etc. Some people distinguish further between contractions (Dr) and abbreviations (Prof), claiming that only the latter should have a period. BE tends to leave them out, AE uses them more often. Above all, be consistent.

Here's what the Chicago Manual of Style has to say:

1. Use periods with abbreviations that end in a lowercase letter: p. (page), vol., e.g., i.e., etc., a.k.a., a.m., p.m., Ms., Dr., et al. (et is not an abbreviation; al. is). An exception may be made for the few academic degrees that end in a lowercase letter (e.g., DLitt, DMin);

2. Use periods for initials standing for given names: E. B. White; do not use periods for an entire name replaced by initials: JFK.

3. Use no periods with abbreviations that appear in full capitals, whether two letters or more and even if lowercase letters appear within the abbreviation: VP, CEO, MA, MD, PhD, UK, US, NY, IL (but see rule 4).

4. In publications using traditional state abbreviations, use periods to abbreviate United States and its states and territories: U.S., N.Y., Ill. Note, however, that Chicago recommends using the twoletter postal codes (and therefore US) wherever abbreviations are used;

Note that the British and the French (among others) omit periods from contractions (Dr, assn, Mme).

No space is left between the letters of initialisms and acronyms, whether lowercase or in capitals. Space is usually left between abbreviated words, unless an abbreviated word is used in combination with a single-letter abbreviation.

RN
C-SPAN
YMCA
Gov. Gen.
Mng. Ed.
Dist. Atty.

but

S.Dak.
S.Sgt.

"MD" ist aber meines Wissens nach die Abkürzung für Medical Director, Medical Doctor wäre doch etwas doppelt gemoppelt.

*f5* #2 is simply wrong, sorry.



Was that first paragraph in #1 by any chance a quote from me in a past LEO thread? It sounded familiar ...

Chicago is a reasonable guideline if you want to follow a manual, but just be aware that you don't have to follow any manual slavishly, and that different people may follow different style guides. Abbreviations in particular are an area of usage that has been changing in recent decades and is still in flux, so that you may see different styles all still in use to some degree. As most LEO users are aware, English doesn't have an official Rechtschreibreform that defines a particular date on which everyone changes at the same time; our changes in style and usage happen gradually, by consensus.

M. D. - This is the oldest / most traditional version, but it was difficult to use with computers and word processors, because the space needed to be a nonbreaking space (ideally, a thin nonbreaking space) so that the line wouldn't break between the two parts of the abbreviation. So it fell out of use, to the point where it would now probably look strange to most people. I think you can safely discard it as a current option.
M.D. - This is a semi-traditional compromise that came to be widely used after the advent of computers and word processors, to avoid the linebreak problem. It's still used by many people, but to many others it may look unnecessarily conservative or old-fashioned.
MD - This is the most modern and most common version. Over time, many abbreviations that were once commonly written with periods have now come to be commonly written without them.

Chicago's latest edition (is there a 17th?) has evidently departed considerably from other relatively recent ones, which just goes to show how quickly style preferences can change in the internet and computer era. It does offer a more simplified reasoning behind the various rules for periods in abbreviations. However, it may have oversimplified to the point of being unreliable in a couple of places, if Carullus has cited correctly.

For instance, S. Dak. (South Dakota) is indeed an older-style abbreviation, but when it is used, it absolutely requires a space between the two parts of the abbreviation, because it is two words. That is, this supposed exception 'unless an abbreviated word is used in combination with a single-letter abbreviation' is total garbage; they seem to have just made that up out of thin air, so I would expect a lot of people to protest and the error to be corrected in the next edition.

Similarly with S. Sgt. (Staff Sergeant); military usage may indeed omit the space, and probably also the periods, compressing it to SSgt, which looks more like a four-letter code than a title. But normal usage in other contexts does not have to follow military conventions, which is good, because military abbreviations and acronyms tend to cross the line into jargon.

Also, two-letter as an adjective absolutely requires a hyphen.

Thank you very much!

hm -- us has given us the typographical low-down, but it's worth pointing out that MD (however spaced) is not the abbreviation of 'Medical Doctor' but of the academic degree of 'medicinae doctor', which in Britain at least is a so-called 'higher doctorate' which very few people have. And in Oxford the abbreviation is DM.

Most British doctors are BM, or BCh, or both. (Sometimes written MB or ChB or BChir or ChirB.)

In AE the vast majority of doctors are MDs; whether you explain the Latin as 'medical doctor' or 'doctor of medicine' or what is up to you, as no one uses the long version anyway. But it doesn't mean 'director' (at least in AE), and it's not meaningless; there are also doctors of dentistry (DDS), osteopathy, and so on, not to mention the many people called 'Dr.' who are PhDs, that is, non-medical doctors, holders of academic doctorates.

A higher medical doctorate in AE is typically an MD/PhD, that is, a medical degree plus an additional academic research degree.

Was that first paragraph in #1 by any chance a quote from me in a past LEO thread? It sounded familiar ...

I'm sure you have said something along these lines, but this paragraph was in fact written by me and is not simply an unattributed quote :)

... if Carullus has cited correctly.

It's a verbatim quote; I've only left out a few cross references.