I strongly agree with those that already suggested to use the standard abbreviations (used in MathSciNet and Zentralblatt Math.), which is what various mathematical journals explicitly suggest/requier (I believe it is not always enforced though).
Some (additional) reasons.
a. While it is save to assume that every human wanting to read your paper will understand what "Trans. of AMS" means, this is much less clear for "non-human readers" who might only know the official abbreviation "Trans. Amer. Math. Soc.".
For example in MathSciNet lists of references are (in part) included and cross-linked;
also some journals do so, and then there are providers of citation statistics (which, as long as they exist and are used, should at least be accurate). It is my understanding that all this is done automatically, to the extent possible. The usage of various different abbreviations makes this task significantly harder.
b. As said for your concrete example "Trans. of AMS" I think there is little risk of confusion.
Even, if you take this one step further and just write "TAMS", chance are people will understand. However, for slightly less universially known journals; things start to become less clear, and even then it can be confusing. It once took me a while to figure out that "CPC" refers to "Combinatorics, Probability and Computing" with the official abbreviation "Combin. Probab. Comput." I would not have had any problem.
(And, when I had that problem, I was very well aware of the existence of this journal, it was just I could not link it instantly to "CPC".)
c. Why, then, not full names? I have no strong opinion against that. But:
i. it is not what journals ask for.
ii. they can be really long.
iii. some journals are not so well-known under their full name.
Again a personal anecdote: In a list of journals (all kinds of subjects, but an incomplete list) I was searching for the journal "Combinatorica", what I found was the full name "Combinatorica. An International Journal on Combinatorics and the Theory of Computing".
Before comparing the ISSN-number I was uncertain whether this is the journal I am looking for, or something else. (Well, I could and should have known the full name, but I never noticed it before.)
d. A journal that is very well-known now, and thus will be recognized by a short ad-hoc abbreviation, can stop to exist or be renamed.
Then, perhaps, in several decades (and this is not an unreasonably long time for a mathematical paper) this journal will be not well-known anymore, and an informal abbreviation will cause trouble. (With an official one, one can always look it up.)
e. Some journals are only very slightly renamed, different series and alike. Even worse, different journals have similar names. Or, conversely, while the journal is renamed the abbreviation is kept the same (for continuity).
If one is not careful one might miss these small difference when making up ones own abbreviations.
To give an example:
'Archiv der Mathematik' is officially abbreviated 'Arch. Math. (Basel)'.
'Archivum Mathematicum (Brno)' is officially abbreviated 'Arch. Math. (Brno)'.
In case one should only know one of them 'Arch. Math.' might seem the natural abbreviation for either of them.
(The later is also an example for another thing mentioned in this point and also c.: the name I gave was the official (long) name from 88-90; before that it was 'Spisy Přírodovědecké Fakulty University J. E. Purkyně v Brně. Archivum Mathematicum' and now it is 'Universitatis Masarykianae Brunensis. Facultas Scientiarum Naturalium. Archivum Mathematicum')
So, in short, I also suggest to use the standard abbreviations.
Regarding the arXiv links: http://mathoverflow.net/questions/53920/arxiv-number-and-mathscinet is somewhat related (not the question itself, but the subsequent discussion).