When you talk about illustrations style for children’s book, there can be many styles of illustration. Broadly the illustrations are categorized in two ways, one is the hand drawn illustration and the other is the digital style of illustration. The hand drawn illustrations are created with hands and the other style is created in software. If you ask about the popularity of the styles, I will say both the styles are equally popular. The hand drawn brings more reality into pictures with a classic approach where as the digital style of illustration gives more colors and life to pictures. Kids love both styles equally.
There can be further classification of styles under both the categories and these are realistic, semi realistic, whimsical, cartoon, black and white etc. When you talk about these classification, it generally means the type of characters and background used in the pictures. Every story has a different need and based in the type of story, the illustrations must be selected. In some stories, it is better to choose a digital illustration with semi realistic or whimsical characters where as another story may look better with hand drawn illustrations with realistic characters, more like classic illustration style. When there are more animal characters, it is better to use a digital illustration style as it can provide more lively and colorful appeal to the illustrations.
While choosing any illustrator you must also be confident about the style of illustration of that particular illustrator. If possible ask for all the different styles that he/she can create. Also, you must discuss the price for the illustrations and how the payment will be made. Also, it is advisable to sign an agreement and discuss points like ownership of copyright etc.
It is very important to choose a good illustrator for your story and you must keep in mind that you don’t always have to go for the popular illustrator or expensive illustrators, you should choose the illustrator based in his/her skills and suitability for your story.