Within the admin area select the 'themes' menu - and the relevant theme you are using.

Then from the left nav menu, under "Catalog Pages" select 'product details'

Click on "Colors, Fonts, Layout" to expand that menu

Click on "availability" to expand that item

Change the "display" drop-down box to "no"

Select 'save' at top of top.

Repeat process for other pages where the 'availability' is currently being displayed. i.e. category page etc.

Cheers