I've been particularly interested in some of the material concerning Seuss's verse. (You can also read about it at the Dr. Seuss page at Wikipedia; just scroll down to the section called Poetic Meters.) I've been a fan of Dr. Seuss since I was first learned to read -- which, I'm afraid, wasn't that long after he started writing! One of my favorites was The 500 Hats of Bartholomew Cubbins. (Ironically, that book is in prose, not verse.)
You often hear it said that publishers don't want stories in verse. That's not exactly true. They just don't want bad stories in verse. Of course, there are some publishers who won't look at any kind of verse... but since it's extremely easy to self-publish your own work, that's not really a problem anymore.
Still, writing good verse can be a problem. I have a poetry blog called Will Shakespeare for Hire where I'm an out-of-work Will Shakespeare trying to recreate himself as a web poet. Poor Will will write anything, from serious pieces to parodies of nursery rhymes to Viagra ads, as long as they're in verse. And sometimes Will deliberately writes bad verse, just for fun. So I guess that makes me, if not an expert, at least experienced in writing verse... both bad and otherwise.
Part of the craft of writing easy reader verse -- and it's the easy reader aspect that this blog is interested in -- is making the verse both easy to read and fun to read. When Seuss wrote The Cat in the Hat, he expected to take a few weeks. Instead, it took him nearly an entire year!
Of course, Seuss was writing from a strict word list, which isn't as important these days. But readability is still important. You don't want to fill your lines with strings of connector words like of, or, and, etc., even though those words have a use in your verses. And since most easy reader verse will rhyme -- it helps the kids read the verse more easily, since the rhymes give them a clue to the sounds of the words -- you need to choose good rhymes. Try reading Horton Hears a Who! and look at some of the rhymes. Although you'll find lots of simple rhymes, you'll also find rhymes like:
- air and Mayor
- working and shirking
- roar, more, and floor
- nowhere and despair
- chirp and twerp (I particularly like this one!)
I'll probably come back to "the verse question" in the future but if you really want to write stories in rhyme, go right ahead. Just make sure you put the time into it and do it right. The Wikipedia article can give you some help about how to structure humorous verse.