Use "ice ax" as the subject of your sentence when you are describing what an ice ax is or does. For example, "My ice ax kept me from plummeting into the black abyss below."
Use "ice ax" with "of" when you are discussing a characteristic or a part of an ice ax. For example, "The head of this ice ax is made of a priceless steel alloy from Portugal."
Use "ice ax" as a direct object in your sentence when you are describing something that is done with an ice ax. For example, "I swung my ice ax into the friable gray snow above my head."
Use "ice ax" as an indirect object in your sentence when you are describing something that is done to an ice ax. For example, "I put all my trust in my ice ax."