Gerunds and present participles can confuse users since both end with -ing.
In your sentences, the first one could be either one. If it's without the apostrophe, the -ing form is a present participle, which acts as modifier to the preceding noun (Jamie), similar to an adjective. So, "I heard Jamie" who is singing. If an apostrophe appears, the -ing form is a gerund, which is a noun form, and it becomes the object instead of Jamie: "I heard singing" As it happens the singing belongs to Jamie. It is true that the latter seems more formal.
In the second sentence, it's the same really. The choice of usage depends on the idea you want to provide and maybe it could be slightly more emphasis on the who or on the what.
When you have pronouns such as "someone," the tendency is to not use the possessive gerund.