To add to what mij59 has written,
If you try to measure resistance or continuity using a multimeter on any one side (i.e primary or secondary) , you will ALWAYS read a dead short.
In case of transformers, the only short you should worry about is between primary to secondary. If you read a short between any winding on the secondary to any winding on the primary, you have a problem.
However, transformers are VERY robust as mentioned. In all my 6 years of working in power electronics industry, The only time I have seen a transformer get damaged is due to a fire.