Try using below endpoint:
http://<sitecollection>/<site>/_api/web/lists/getbytitle('Display name of List')/contenttypes
OR
http://<sitecollection>/<site>/_api/web/lists(listid)/contenttypes
official documentation: SP.ContentTypeCollection object - REST resource endpoint.
Update:
Try using something like below to fetch the names of content type. This is like pseudo code for your reference. You can check the exact data you are getting from REST call by adding break-point inside success
function:
$.ajax({
url: _spPageContextInfo.webAbsoluteUrl + "/_api/web/Lists/getbytitle('Display name of List')/contenttypes?$select=Name,Id",
type: "GET",
headers: {
"Content-Type": "application/json;odata=verbose"
"Accept": "application/json;odata=verbose"
},
success: function(data) {
var optionsArray = [];
if(data.d && data.d.results) {
for(var i = 0; i < data.d.results.length; i++) {
console.log(data.d.results[i].Name);
// Push the content type name in array
optionsArray.push(data.d.results[i].Name);
}
}
},
error: function(error) {
console.log(error);
}
});