Nobody warns you about the hundredth reading.
You buy a book, the child likes it, and then you read it every night for four months. By the end you know whether the rhyme was any good, because bad rhyme becomes unbearable at around reading twenty and good rhyme somehow does not.
Here is what separates them, and a few that hold up.
What makes rhyme survive
The stress falls where the words naturally fall. Most bad children's rhyme is technically rhyming and rhythmically wrong, so you have to squash a word to make it fit. Once you notice it, you notice it every night forever.
The rhyme is not the joke. If the entire pleasure is that two words sound alike, it dies fast. If the rhyme is carrying a story, it lasts, because the story is still happening.
There is something for the adult. The best ones have a line or two aimed over the child's head, and that is not indulgence, it is what keeps the reader alive at reading sixty.
The ones that hold up
Julia Donaldson, most of them. The Gruffalo and Room on the Broom are the famous two, and the reason they survive is that the rhythm is genuinely correct. You can read them fast without stumbling, which after four months matters enormously.
Dr. Seuss, but not all of it. Green Eggs and Ham survives anything. Some of the longer ones do not, because the rhyme starts driving the story instead of the other way round.
Each Peach Pear Plum. Short, quiet, and built on a game rather than a plot, so it works at two and still works at five for different reasons.
Almost any traditional nursery rhyme collection. These have been tested by a few hundred years of parents, which is a filter no publisher can match. If a rhyme has survived since 1780 it will survive your March.
What to avoid
Books where the rhyme has been added to an idea, rather than the idea arriving in rhyme. You can hear it immediately. The sentences are the wrong shape, and there is usually an unnecessary word at the end of a line doing nothing but rhyming.
Also avoid anything where the moral is in the last two lines in a different voice. Children notice the change of tone and it tells them the story was a wrapper.
Where mine sits
Sing Me a Fable is twenty-six fables in four verses each, and I will tell you the design decision that came from exactly this problem: the moral is not appended at the end in a solemn voice. The fault is named at the start, in gold, and then the story simply happens and stops. I wrote about why.
Its honest weakness for this particular list: at four verses, a fable is longer than a two-year-old will sit through, so it is a four-to-eight book rather than a toddler one.
Its strength is the same thing that makes the traditional collections survive. Twenty-six of them means you are not reading the same one a hundred times, you are reading twenty-six of them four times each, which is a considerably kinder arrangement for the adult.
Or start free: twenty-six colouring posters, one per letter, no charge.