I thought there was a thread like this already, but I can’t find it. Here’s the latest one I heard:
剪刀石頭布
你沒穿內褲
內褲加在聯絡簿
送給老師當禮物
老師看了很生氣
一腳把你踢到美國去
美國小姐真美麗
送你一盒巧克力
邊走邊去邊方屁
阿里爸爸打屁屁
Wrote it down in pinyin, so I’m not sure if I got all the characters right, but it should be pretty close.
jiandao shitou bu
ni mei chuan nei ku
song gei laoshi dang liwu
laoshi kan le hen shengqi
yi jiao ba ni ti dao meiguo qu
meiguo xiaojie zhen meili
song ni yi he qiaokeli
bian zou bian qu bian fang pi
a’libaba da pipi
(Lose translation)
Paper, Rock, Scissors
You’re not wearing any underwear
You gave them to your teacher as a present
The teacher saw it and got mad
She kicked you to America
American girls are really pretty
They gave you a box of chocolates
You walked here and there farting
Alibaba spanked you
Now everyone can enjoy 
(It rhymes in Chinese of course)
That is funny! My son has all sorts of those rhymes he brings home from school.
The only correction would be:
Jiandao shi tou bu is:
scissors rock cloth (but you knew that of course)
Two-oo tigers
Two-oo tigers
Running fast
Running fast
One has got no eyes
One has got no tail
Very strange
Very strange
(to the tune of Frere Jacques)
兩隻老虎
兩隻老虎
跑得快
跑得快
一隻沒有眼睛
一隻沒有耳朵
真奇怪
真奇怪
I think they teach that at Shida, but I have heard them sing it in school.
Here is one I just found while digging around in my Chinese - English dictionary software directory:
一二三四五六七
鬼子上山來抓雞
抓了雞還不夠
還讓老頭下倆蛋
老頭氣得沒法子
坐在炕上數麻子
My favorite Taiwanese rhyme
ku5 to7si7 ku5
khan1 ka3 pak4-kia*1 ma7si7 ku5
牛就是牛,
牽到北京還是牛
roughly translated into German: dbddhkp
Doof bleibt doof, da helfen keine Pillen
Don’t know about any English equivalents but there must be loads.