How to get Ajax posted Array in my C# controller?

Set the contentType: "application/json" option and JSON.stringify your parameter:

var lines = new Array();
lines.push("ABC");
lines.push("DEF");
lines.push("GHI");
$.ajax(
{
    url: 'MyController/MyAction/',
    type: 'POST',
    data: JSON.stringify({ 'lines': lines }),
    dataType: 'json',
    contentType: 'application/json',
    async: false,
    success: function (data) {
        console.log(data);
    }
});

You can also set the type of objects you're getting if it makes sense in your business case. Example:

public JsonResult MyAction(string[] lines)
{
    Console.WriteLine(lines); // Display nothing
    return Json(new { data = 0 });
}

And, something more practical with what you're sending in:

public class MyModel {
    string[] lines;
}

and finally:

public JsonResult MyAction(MyModel request)
{
    Console.WriteLine(string.Join(", ", request.lines)); // Display nothing
    return Json(new { data = 0 });
}