You currently cannot change which badges display on your Profile or Dashboard. Here's what's displayed on each:
Dashboard - The most recent Writing Badge and Special Badges you've earned will display on your dashboard, starting with the most recent. Challenge badges do not display on the dashboard. Only the top badge of each category is shown (so, if you earned two Author badges in a row, only the highest ranking Author Badge would be shown).
Public Profile - The most recent Special Badges you've earned will display on the "Recent" tab of your public profile. As on your Dashboard, only the top badge of each category is shown. Secret Badges and Challenge Badges are not included in this display, but they are available publicly on your Badges tab.
Public avatar "hover card" - Same as Public Profile above.