Should I validate transactions using the rules in BIP62? BIP62 introduces a number of rules to prevent transaction malleability. However many transactions already in the blockchain fail these tests. Is BIP62 currently being used to filter transactions from entering the blockchain? If not then are there any plans by the core development team to use BIP62 like this?
Yes, you should.

BIP62 is currently not being pursued as a consensus rule on the network.

However, all its rules (including the ones that are only applied to new style transactions according to BIP62, specifically the lower-S rule) - are currently being enforced as a standardness rule since Bitcoin Core 0.11.1 and 0.10.3.

The reason for this is that at the time BIP62 was written, requiring lower-S was not feasible due to the large amount of wallets that were still creating high-S signatures. In recent times, a large fraction of them switched to lower-S, making it viable to just unconditionally apply that rule. If a significant amount of network nodes (and specifically, miners) upgrade to versions that enforce this, wallets that do not enforce lower-S will gradually stop working and need upgrading.

If that process completes and we end up in a state where near-0% of blocks still contain high-S signatures, people may want to propose unconditionally applying it as a consensus rule.