Schools of thought (madhabs)
Shafi'i inheritance
Followed in Egypt, the Levant, and Southeast Asia. When the grandfather inherits with siblings he gets the best of three options, and the school applies the Musharakah ruling.
The Shafi'i school dominates Egypt, the Levant, and Southeast Asia (Indonesia, Malaysia). Its inheritance rulings are very close to Maliki and Hanbali, and on the contested special cases it disagrees with Hanafi.
Where Shafi'i differs
Applies Musharakah
Where Hanafi gives full siblings nothing in the Musharakah case, Shafi'i has them share with the maternal half-siblings in the 1/3 portion, even though they would otherwise be residuary heirs with nothing left to claim. The reasoning is that they share the same mother, so it's fair to include them.
The grandfather with siblings
On the grandfather-with-siblings case, Shafi'i computes three options for the grandfather (1/3 of remainder, share-as-sibling, 1/6 minimum) and gives him the largest. Maliki and Hanbali use the same method, with small differences in the sub-rules.
Applies Radd
Shafi'i has historically been more accepting of Radd than Maliki, and applies it readily in modern practice.