These are interesting questions at the moment!
The attached is an analysis of the "no palm oil" claim - it's from TheOilPalm.org, so is perhaps slightly biased (the site is run by the Malaysian Palm Oil Council), but it contains some interesting discussion in more detail than I can sensible type here. There certainly are still plenty of products making this claim.
The "no artificial flavours" claim is something I still see on various labels here, but I'm somewhat sceptical as to the actual legal basis for it. Under the current EU flavouring regulation, (EC) 1334/2008, there is no longer an "artificial flavours" category. Thus it could be argued to be the case that no product on sale in the EU contains an artificial flavour, and so any such claim would theoretically fall foul of Article 7(1)© of Regulation (EU) 1169/2011: Food information shall not be misleading, particularly: by suggesting that the food possesses special characteristics when in fact all similar foods possess such characteristics, in particular by specifically emphasising the presence or absence of certain ingredients and/or nutrients.
i.e. if no food on sale in the EU contains an artificial flavour, since the concept doesn't exist in current flavouring regulation, then all food meets the requirement of this claim and so it is misleading to use it because it suggests the food is special when it is not.
Alas it is not that cut and dry though, as one could also argue that there could be a legitimate consumer understanding in terms of an artificial flavouring implying that the flavour has been derived from synthesis in a laboratory or similar, rather than produced from naturally-occuring substances. To my mind this is tenuous, but some regulators seem to be turning a blind eye, if not outright endorsing it.
You could of course look at turning it around into a positive claim instead, as for example "made with natural flavours" would be legitimate, assuming the flavourings you are using meet the required definition in Regulation (EC) 1334/2008.
The flavour enhancer claim is going to require a bit more reading/information. For example, I couldn't make this claim about a carton of apple juice because flavour enhancers are not permitted in any pure fruit juices, and thus all fruit juices on sale in the EU will contain no flavour enhancers so the claim doesn't comply with Article 7(1)© of Regulation (EU) 1169/2011, as discussed above.