prepend.rb

s = 's';
def s.foo; 'foo'; end

eigen = (class << s; self; end)
def eigen.foo; 'bar'; end   # This is unused

def String.foo; 'baz'; end

module Frob
  def foo
    super+'frob'
  end
end

p s.foo
#=> "foo"
class << s; prepend Frob; end
p s.foo
#=> "foofrob"

p s.class.foo
#=> "baz"
class << String; prepend Frob; end
p s.class.foo
#=> "bazfrob"

prepend_class.rb

module B
  def a
    puts 'b'
    super
  end
end

class A
  class << self; prepend B; end
  def self.a
    puts 'a'
  end
end

class << A
  def a
    puts 'c'
    super
  end
end

A.a

# =>
# b
# c
# prepend_class.rb:18:in `a': super: no superclass method `a' for A:Class (NoMethodError)
# 	from prepend_class.rb:4:in `a'
# 	from prepend_class.rb:22:in `<main>'
#
# I'm assuming this is because the singleton class is the class, so it redefines the method rather than overriding...
#
#

prepend_instance.rb

module B
  def a
    puts 'b'
    super
  end
end

class A
  prepend B
  def a
    puts 'a'
  end
end

a = A.new
class << a
  def a
    puts 'c'
    super
  end
end

a.a

# =>
# c
# b
# a