I've been thinking about this subject recently too, as my Sunny can't have NSAIDs as he had a bad reaction (staggering around, loss of coordination) from just one injection.
When Sunny had a recent problem with limping, instead of a NSAID we had Tramadol for pain relief on an as-needed basis but as others have said this does not have the anti-inflammatory properties of the NSAIDs.
There are also herbal anti-inflammatories that can be given to dogs, such as Devils Claw. I've recently bought one for Sunny that's a combination of Devils Claw and Yucca but have yet to use it so don't know how effective it is. You also have to bear in mind that herbal remedies can also have side effects, such as gastric disturbance.
Re steroids, because of the serious side effects, I think I would want to reserve these for very serious conditions only.