My MVP for catching Pokémon was Akira, my Mightyena. She could withstand a lot of hits while not unintentionally defeating the wild Pokémon. She was definitely my go-to Pokémon, if things went badly she always persevered for me.
For battles my MVP was most certainly Casimira, my Absol. With just one Swords Dance she was able to sweep most teams. Sadly though her defences were lacking and her speed was average, so she was out sped a lot. She had a lot of trouble at the League in particular, but she pulled through and even defeated Steven's Mega Metagross!
In the overworld of Eastern Hoenn, Scylla, my Sharpedo was definitely my MVP. She was able to take me everywhere I wanted, and rapidly too. In battles she was no weakling either, in fact it was thanks to her that I was able to defeat Glacia where all my Pokémon fell to Freeze-Dry and Blizzard, she came out on top.
In general, Shido, my Shiftry was well rounded in every area. He was my backup force, but he was fragile. While he was still a Nuzleaf he couldn't take any one hit without fainting, but he grew stronger. Once he evolved into Shiftry, he too became a force to be reckoned with, especially with his Leaf Storm.
My team were all my MVPs, I can't say it any other way. They all had their strengths and weaknesses, but they all covered for one another.