Replace each whitespace in a string with “%20”

This one really fits well for a regular expression:

public static string ReplaceAllSpaces(string str) {
  return Regex.Replace(str, @"\s+","%20");
}

The expression \s+ consists of the pattern \s that matches a whitespace character (which is more that just spaces, as noted by @tinstaafl), and the quantifier + which means "at least once".

The pattern matches "one or more whitespace characters", so it will match each group of whitespace characters (for example the six spaces between My and Name in the example) and replace it with %20.

Since a string is basically a character array, I would do away with the stringreader and simply loop through the string itself:

    static string ReplaceAllSpaces(string input)
    {
        StringBuilder builder = new StringBuilder();
        bool continuousSpace = false;
        foreach (char c in input)
        {
            if (!continuousSpace && char.IsWhiteSpace(c))
            {
                builder.Append("%20");
                continuousSpace = true;
            }
            else if(!char.IsWhiteSpace(c))
            {
                builder.Append(c);
                continuousSpace = false;
            }
        }
        return builder.ToString();
    }

A side note: Whitespace includes a lot of other characters in the unicode set as well as the space.

White space characters are the following Unicode characters:

Members of the SpaceSeparator category, which includes the characters

SPACE (U+0020), OGHAM SPACE MARK (U+1680), MONGOLIAN VOWEL SEPARATOR (U+180E), EN QUAD (U+2000), EM QUAD (U+2001), EN SPACE (U+2002), EM SPACE (U+2003), THREE-PER-EM SPACE (U+2004), FOUR-PER-EM SPACE (U+2005), SIX-PER-EM SPACE (U+2006), FIGURE SPACE (U+2007), PUNCTUATION SPACE (U+2008), THIN SPACE (U+2009), HAIR SPACE (U+200A), NARROW NO-BREAK SPACE (U+202F), MEDIUM MATHEMATICAL SPACE (U+205F), and IDEOGRAPHIC SPACE (U+3000).

Members of the LineSeparator category, which consists solely of the LINE SEPARATOR character (U+2028).

Members of the ParagraphSeparator category, which consists solely of the PARAGRAPH SEPARATOR character (U+2029).

The characters CHARACTER TABULATION (U+0009), LINE FEED (U+000A), LINE TABULATION (U+000B), FORM FEED (U+000C), CARRIAGE RETURN (U+000D), NEXT LINE (U+0085), and NO-BREAK SPACE (U+00A0).

Is there anything wrong with this?

string replaceAllSpaces(string input)
{
    return input.Replace(@" ", @"%20");
}

Maybe add some input validation, if you need.

Since it is only one specific char we want to replace, there is no need to whip out a Regex, especially if you're not comfortable around those.
On the other side the Regex approach is better in terms of scalability, if you ever need to complicate the criteria for a replacement.

Also since only ASCII-chars are allowed, there is not problem with the dozens of other chars that are practically whitespace but not the whitespace, that are included in unicode. (See tinstaafl's answer).