Javascript, Razor and Escape characters. Like apostrophe

HttpUtility.JavaScriptStringEncode is not really required here. Simply

 '@Html.Raw(s.Name)'

is worked for me.

Try like this:

$(function () {        
    $('#calendar').fullCalendar({
        header: {
            left: '',
            center: 'title',
            right: 'month,agendaWeek,agendaDay'
        },
        month: 5,
        year: 2011,
        editable: false,
        events: @Html.Raw(new JavaScriptSerializer().Serialize(ViewBag.Sessions))
    });
});

ViewBag.Sessions might require some modifications to achieve the desired result (in terms of property names), which brings me to the usual remark I make about ViewBag when I see someone using it: using ViewBag is bad practice and I would recommend you using a strongly typed view with a view model.

I would write your foreach like this:

            @foreach (var s in ViewBag.Sessions)
            { 
                <text>
                {
                 title: '@HttpUtility.JavaScriptStringEncode(s.Name)',
                 start: new Date(@s.Starts.Year, @s.Starts.Month-1, @s.Starts.Day),
                 end: new Date(@s.Ends.Year, @s.Ends.Month-1, @s.Ends.Day)
                },
                </text>
            }

• HttpUtility.JavaScriptStringEncode to escape quotes and html markup.
• <text> is nicer for multiline output.

Here is how to do it:

title: '@Html.Raw(HttpUtility.JavaScriptStringEncode(s.Name))'