This is going to sound crazy. It's from my mother-in-law but she passed it down to us. She was a very young and inexperienced mother when she had her 1st two. When life would get too much for her she'd calmly sing this to keep sane (to the toon of Row, Row, Row Your Boat):

Beat, beat, beat the children
Beat them everyday
'cause if you don't beat them everyday
they grow up mean and nasty

This would be repeated for hug the children and love the children. By the time she got through the last verse she'd usually feel in control again.

Amazing Diaper (Yes this is somehow wrong, but the kids love it)

Amazing Diaper, how much it holds

when baby needs to pee

It once was dry, but now is soaked

Who gets to change him? I guess that's me

Amazing Diaper, how much it smells

When baby has to poo

The changing task requires a gas mask

And a diaper genie too

One of my made up songs for our daughter is: ( to the tune of "Are you Sleeping")

We love Maelie, We love Maelie Yes, we do, Yes, we do, we love her when she's sleeping, we love her when she's scrunchy, yes, we do, yes, we do.

I add verses with different things that she does, and different emotions..... eating, excited, happy, hopping, jumping, etc....

I'll join in:

Creamy bot
Creamy bot
Creamy in the morning
Creamy in the noon-time
Creamy bot
Creamy bot
Creamy when the son goes down

to the tune of Praise Him

Sung to a child with nappy rash.

To get things started, I'll answer my own question.

To the tune of Frosty the Snowman:

I've got a nappy, I've got a terry square,
I can wear it on my bottom, I can wave it in the air.
I've got a nappy, I've got a terry square,
It's lovely, soft and fur-ry, just like my teddy bear.

(Note to Americans: "nappy" = "diaper")

Since my oldest is named Alice, some members of my family like to sing

Alice where are you going?
Upstairs to take a bath.
Alice has legs like toothpicks
and a neck like a giraffe-affe-affe-affe
Alice pulled out the plug
Oh my goodness, oh my soul
There goes Alice down the hole
Alice where are you going?
Glub glub glub

I refuse to sing it to her because I don't want to make her scared of baths so I do

Alice the camel has 2 humps
Alice the camel has 2 humps
Alice the camel has 2 humps
So roll, Alice, roll

Alice the camel has 1 hump
Alice the camel has 1 hump
Alice the camel has 1 hump
So roll, Alice, roll

Alice the camel has no humps
Alice the camel has no humps
Alice the camel has no humps
So Alice is a horse.

I know some silly songs however, they are from various books. These are little poems that help create loving and soothing atmosphere for the child when is time to sleep. Here is one for a small baby:

"Eenie meenie inka-boo, Crunka runka, bedtime for you. Onka wonka bliddly bleep Minka, pinka, go to sleep."

Another:

NIghty-night, turn out the light, Its' time for bed. Lay down your head. Here come two arms that will hug you real tight. Here is your mom/dad who will kiss you good night.

Another: You sing it to the tune of "Twinkle, Twinkle"

Soon the moon at you will peep. Now it's time to go to sleep. Stars will shine way overhead, Making sure you're safe in bed. Down a mountain very steep. Meet me in the land of sleep.

Another:

This is my pinkie, and this is my thumb. I'm getting in bed, and I'm counting to one. These are my eyes. Are they brown? Are they blue? I'm lying down snug, and I'm counting to two. These are my feet, and I'm wiggling them free. I'm stretching and settling and counting to three. These are my ears. I have one , and one more. I'm rubbing the sheet, and I'm counting to four. These are my hands;under covers they dive. I'm comfy in bed, and I'm counting to five. I've counted to five, so now turn out the light. My bed feels so cozy. I love you. Good night.

These rhymes are from a book Night-Night: Settle Down Activities for Easy Bedtimes. By Cynthia Mac Gregor

Another wonderful book about lots of lovely things to do with your child is "I love you Rituals" by Becky A. Bailey