For medical residency:
* Internship (specifically for the first year)
* Training
* Fellowship (for post-residency specialization)
* Postgraduate medical education
* Clinical training
For other types of residency:
* Live-in
* Permanent residence
* Dwelling
* Home (in informal contexts)
* Domicile (legal term)
* Stay (temporary residency)
* Sojourn (temporary residency)
To be more specific, please tell me:
* What kind of residency are you referring to? (medical, legal, etc.)
* In what context is the word being used?
Once I have this information, I can give you a more precise synonym.