Only walk-up fares, and only some of them at that, are regulated:
- For short-distance journeys, usually it's the Anytime Day Return that is regulated.
- For medium-distance journeys that have both day returns as well as period returns (i.e. valid for return within a month), the Anytime Day Return and Off-Peak Return are typically regulated.
- For long-distance journeys that don't have day returns, usually it's the Off-Peak Return that is regulated.
For some operators (e.g. GWR, EMR and LNER for journeys to/from London and the southeast) it's the Super Off-Peak rather than Off-Peak Return that is regulated. In essence it's whichever fare is deemed the successor to what was the Saver Return in 2003 - before the fares "simplification" exercise conducted in 2008. In a handful of cases (usually long-distance journeys priced by regional operators such as Northern) this is instead an Anytime Return or Anytime Short Distance Return.
The cost of weekly (and longer) season tickets is usually also regulated, where they exist. And in the London Zones, Oyster/contactless PAYG peak fares are regulated instead of single or return paper/smartcard tickets.
Advance fares aren't directly regulated by the government. That being said, the ORR did enter into an agreement with Abellio regarding Advance pricing when they took over the EMR franchise - there were concerns about a reduction in competition as Abellio now operated all services on the Peterborough to Norwich axis as they also held the Greater Anglia franchise.
Furthemore, as stated in my post of 2021 that
@jfollows kindly linked to, the notion of 'regulated fares' is somewhat illusory under the current environment. TOCs are either directly owned and operated by the government (e.g. Northern and LNER), or are under a National Rail Contract that obliges them to do whatever the government says (the 'management contract' arrangements mean the amount of revenue generated makes little difference to operators' profits), or they are Open Access operators to whom fares regulation doesn't apply.
I also wouldn't be surprised if the government were directing TOCs as to the percentage by which unregulated fares - such as Off-Peak Day Returns and Advances - are increased.
The government has also directed TOCs to make several changes that, if they had been undertaken by a TOC pre-Covid, would have been non-compliant with fares regulation. For example, they abolished return fares for most flows priced by LNER and in so doing, sneakily increased prices by around 5% on top of the annual fare increase. Similarly, some Off-Peak Returns have been abolished in London and the southeast as part of the expansion of contactless PAYG.