C# naming convention for constants?

The recommended naming and capitalization convention is to use PascalCasing for constants (Microsoft has a tool named StyleCop that documents all the preferred conventions and can check your source for compliance - though it is a little bit too anally retentive for many people's tastes). e.g.

private const int TheAnswer = 42;

The Pascal capitalization convention is also documented in Microsoft's Framework Design Guidelines.

Visually, Upper Case is the way to go. It is so recognizable that way. For the sake of uniqueness and leaving no chance for guessing, I vote for UPPER_CASE!

const int THE_ANSWER = 42;

Note: The Upper Case will be useful when constants are to be used within the same file at the top of the page and for intellisense purposes; however, if they were to be moved to an independent class, using Upper Case would not make much difference, as an example:

public static class Constant
{
    public static readonly int Cons1 = 1;
    public static readonly int coNs2 = 2;
    public static readonly int cOns3 = 3;
    public static readonly int CONS4 = 4;
}

// Call constants from anywhere
// Since the class has a unique and recognizable name, Upper Case might lose its charm
private void DoSomething(){
var getCons1 = Constant.Cons1;
var getCons2 = Constant.coNs2;
var getCons3 = Constant.cOns3;
var getCons4 = Constant.CONS4;
 }

Actually, it is

private const int TheAnswer = 42;

At least if you look at the .NET library, which IMO is the best way to decide naming conventions - so your code doesn't look out of place.