Psalm 31:1 "In thee, O Lord, do I seek refuge; let me never be put to shame; in thy righteousness deliver me!"

What puts us to shame? Shame is different from embarrassment, and David is not asking to never feel embarrassed, he is asking to never be put to shame. Shame is a feeling of discomfort, sometimes extreme discomfort, from being conscious of one's own bad behavior. We don't feel shame for making mistakes, although we might feel embarrassment, we only feel shame when we know we have chosen to do wrong. That is why confession can be so difficult, we have to confront our shame. It's easy (and not even necessary) to confess accidentally hurting someone, but it's much harder to confess that you purposely hurt someone (in fact, often we justify such behavior as being provoked and don't even admit to ourselves that we have sinned!)

David was, I'm sure, confident enough in himself and his relationship with God to not feel shame unless it was well deserved. So what he is asking God for, then, is protection from falling into sin, which would lead to shame. He later asks God to take him out of "the net that is hidden" for him (Psalm 31:4). That is a good description of the trap of sin. It is hidden, just waiting to ensnare us. The best way to avoid it is to seek refuge in the Lord, and that is exactly what David is doing.

God loves this kind of prayer, He wants to protect us from sin, and thus shame. And there is nothing hidden with God. If we seek refuge in Him as David did, placing ourselves completely in His protection, wanting nothing for ourselves, He will save us from the traps and snares of sin. He will never let us be put to shame the only place it matters, before Him, when we are doing His will.