Дразнилки!
The teases!

Тили-тили-тесто
Жених и невеста!
About a boy and a girl who are suspected to be in love (even if they are not) Something like tili-tili-tide-here go a husband and bride!

У Ивана [or any name in genetive case] на носу
Ели черти колбасу!
Just a tease. Like On the Mr. Jones' nose the devil ate the sausage!

A more rude one:
Иван Иваныч [the name and surname] - дуралей
скушал баночку соплей!
Silly John Pott ate a jar of snot!

About red-haired person:
Рыжий-рыжий, конопатый,
Убил дедушку лопатой!
The first letter of the second line must be stressed only for rythm, otherwise it is wrong stressed.
Somewhere must be a cartoon about this red-red-haired:
Red-red, freckled - killed his grandpa by the spade!

Кто так обзывается,
Тот сам так называется!
When someone is offending you you say:
Who said this are the same they said!

Первое слово съела корова!
Второе словечко съела овечка!
First is when you need to say that the words which were just said do not count at all. But your friend can answer by the second phrase, which means the same about your words.
The cow has eaten the first word!
The sheep ate the last one!

Плакса-вакса-гуталин
На носу горячий блин!
Something like that, though I surelly fantasized a little bit:
Sissy-peessy lose
with the pancake on his nose

Жадина-говядина турецкий барабан
Кто на нём играет - Ванька таракан!
About a greedy person -
Greedy-beef and Turkish drum
Who plays on it& - It's John-the-cockroach!

Кто скажет слово - тот дохлая корова!
A nice one: Who will say a word is a dead cow!

A little of childish fantazy and we have that one:
Шёл по крыше воробей
Нёс он баночку соплей
Кто хоть слово пикнет
Тот её и выпьет!

The sparrow went along the roof
carrying a jar of snot!
If anyone will make a sound
he will drink it!

Чик-трак я в домике!
I love it, seems to be international:
Lock-lock, I'm in the house!

Something I remember from my childhood, the things you just screamed without a reason, which makes everyone laughing aloud. Don't repeat if you are not four-five years old:

Обезьяна Чичичи продавала кирпичи
За верёвку дёрнула и случайно пёрнула!
The Chi-chi-chi monkey sold the bricks
Then it has pulled a rope and suddenly farted!