We tend to use the phrase 'mobile home' for a number of like homes. Be sure to specify 'mobile' or 'manufactured'. You can search the restrictive covenants for neighborhoods by clicking the referenced link or by calling neighborhood code enforcement at (260) 427-1118.
You would have to get a variance from the city to put a manufactured home on a city lot. It is a fairly simple process. Just call down to the zoning board and you can sign up to go to the next meeting. Your home would have to be put on a permanent foundation and I do not believe they would allow a singlewide.