Layout handle for product grid and list view in magento

 Layout handles for product grid and list view in MagentoHello

Using the handles: catalog_category_default, catalog_category_layered and catalogsearch_result_index you should be able to change the template of the products listed.

<catalog_category_default>
    <reference name="product_list">
        <action method="setTemplate">
            <template>MyModule/Availability/catalog/product/list.phtml</template>
        </action>
    </reference>
</catalog_category_default>
<catalog_category_layered>
    <reference name="product_list">
        <action method="setTemplate">
            <template>MyModule/Availability/catalog/product/list.phtml</template>
        </action>
    </reference>
</catalog_category_layered>

But for the search results page you would need to make a slight change 
as the block is not called product_list but search_result_list:

<catalogsearch_result_index>
    <reference name="search_result_list">
        <action method="setTemplate">
            <template>MyModule/Availability/catalog/product/list.phtml</template>
        </action>
    </reference>
</catalogsearch_result_index>