I do nearly all my F/O shooting off a bipod these days. A great deal of this is down to my primary range being one in a steep valley with lack of vehicle access to many firing points. Believe me, the difference between carrying a SEB Neo and a modern bipod is noticeable on this ground.
A key factor is the development of VERY wide footprint bipods - 660mm (26-inches) or wider at low settings. The three I have in particular mind are the Tier-One 'Carbon', Dolphin Gun Co. 'AB', and The Shooting Shed 'Stotteben'. The SEB Joypod is also pretty wide and very stable and some UK F/O competitors use it locked-up (ie set it and shoot without using the joystick facility).
As to using a bipod off a bench, BR rules say a rest is required as I understand it. We do have people shoot BR off bipods in the UK and there have been some very impressive 'aggs' produced on occasions especially in 600 yard comps. The heavy Canadian Rempel is particularly usable in this role. There are problems with bipods in BR though that are not applicable to F-Class. For instance, a modern wide FTR type bipod often has a wider footprint than the bench width, so literally doesn't fit. Bipods don't suit concrete surfaces usually increasing the tendency to jump under recoil and/or move about requiring considerable re-aim for each shot and slowing the rate of fire down even when a piece of carpet or similar is placed on the bench-top. Finally the shooter often has to position him/herself behind the bench centreline not in the designed cut-out section at the rearside making for an unnatural / uncomfortable firing position. Nevertheless, some top GB FTR shooters I know though do all their barrel break-in / load development off the bench on bipods and seem to manage surprisingly well despite heavy loads with 185-200gn bullets.
As a long term FTR competitor who now mostly shoots 'Open', I really don't miss a rest when shooting off grass as I mostly do. When the weather is really bad at Diggle and we're driven to shooting off concrete in our bench equipped covered firing points, I revert to a wide forend and the SEB Mini-Rest. My heaviest recoiling / greatest torque cartridge is the 284 with a 180gn bullet at ~2,850 fps from an 8-twist barrel. A .300WSM shooting 230s at 3,000 fps plus MVs may be another matter entirely of course.
I reviewed the excellent Tier-One 'Carbon' here:
http://www.targetshooter.co.uk/?p=2380
......... and I'll follow up with the AB and yet wider Stotteben models shortly. (Incidentally in the write-up, I said Steve Baardsen (Steveb on the Accurate Shooter forum and now owner/maker of Matrix Bullets) was successfully shooting a 7mm SAUM at Raton in the 2013 US Nationals / F-Class Worlds off a Rempel. Steve has corrected me since - it was a .300 SAUM shooting 210s, more impressive still.)