Rendering three types of events as HTML elements
The simplest refactor would be to simply pull out the common functionality, and pass in the array for each case, as well as a function used to create your div content from the current item in the loop.  Assuming dtInt is global:

function checkDtInt(arr, funcToCreateDiv) {    
    if (dtInt in arr) {
        var arrEventsV1 = arr[dtInt];
        for (var v1 = 0; v1 &lt; arrEventsV1.length; v1++) {
            var travelEvent = arrEventsV1[v1];
            $("#dateCupcake" + i).append(funcToCreateDiv(travelEvent));
        }
    }
 }

checkDtInt(calendarEventDict, function(item) { 
   return "&lt;div  dateId='" + item.Date + "' class='event " + "cal" + item.CalendarId + " " + item.CalendarClassName + "' calendarId='" + item.CalendarId + "' eventDateId='" + item.Id + "' id='" + item.EventId + "'&gt;" + item.DisplayName + "&lt;/div&gt;"
});

checkDtInt(vacationEvents, function(item) { 
   return "&lt;div  dateId='" + item.Date + "' class='event " + "cal1 blueAllDay' calendarId='1' eventDateId='" + item.Id + "' id='" + item.EventId + "'&gt;&lt;img src='/Content/Images/Icons/pawn_glass_" + item.ApprovalIcon + ".png' /&gt;" + item.PersonName + "&lt;/div&gt;"
});

checkDtInt(travelEvents, function(item) { 
   return "&lt;div  calendarId='2' dateId='" + item.Date.ToString("MMM dd, yyyy") + "' class='event " + "cal2" + " blueAllDay' calendarId='2' eventDateId='" + item.Id + "' eventId='" + item.TravelRequest.Id + "'&gt;&lt;img src='/Content/Images/Icons/user1_earth16.png' /&gt;" + item.TravelRequest.Person.LastName + ", " + item.TravelRequest.Person.FirstName.Substring(0, 1) + " visiting " + item.TravelRequest.TechnicalCentre.Name + "&lt;/div&gt; "
});