Do I need to show proof of funds in a local bank account for my Thailand visa application?

Anywhere as long as it is your name. Also having a pension can increase your chance of approval (not a must, but good to have).

Check immigration handbook

The thing is....if the authorities smell something fishy, they're just gonna play it safe and disapprove your visa.

Anywhere - in theory. But the more offshore the money is, the more likely that the consulate might have additional questions / require more formal proof - e.g. statements stamped by the bank (or even notarized). You will likely be fine, just be prepared. Application of "rules" is a big and inconsistent mess among different consulates.

The accounts do not have to be banks that are in the country in which you reside however they will need to be instant access accounts.

Anywhere