I give my FB prints a 1 minute rinse after a 1 minute ilford rapid fix 1+4. No stains. After toning; another rinse, HCA, wash. BTW selenium toner can be replenished practically forever. No need to discard it. Search for Doremus Scudder's post on this.
Since I've been mentioned, I'll chime in on this.
Most of the answers above are correct, but we need to collate and organize a bit
First, my work flow:
Develop - Stop - Fix 1 - Thorough Wash, Dry and Store
----then, on another day I do a toning session-------
Water Soak - Fix 2 - Selenium Tone - Wash Aid - Thorough Wash - Stabilizer - Dry - Store
I use Ilford Rapid Fixer or Hypam diluted 1+9 (or a similar rapid fixer in the "print dilution"). I never have problems with staining when transferring directly from fix 2 to the toner (except once when I inadvertently underfixed - the resultant staining actually helped me diagnose the problem).
Which brings us to staining in the selenium toner. There are two reasons a print will stain in the toner. The first is inadequate fixing. In this case, you can get blotches of stain in areas that were not fully fixed or you can get an overall stain that looks like fogging (possibly with a yellow tinge). This is easy to correct; fix well first. As an aside here, your selenium toning step is a good check for adequate fixation. If you don't get stains like these, you've likely fixed well.
The second cause of staining is using a fixer that is too acid for an immediate transfer to selenium toner without an complete wash in between. This can be solved by changing the second fixer too a less-acid one or simply washing completely before toning. Related to this is staining from inadequate washing between the first and second fix; blotchy staining can result here as well. A "rinse" is not adequate in this scenario; a full wash is needed.
Sometimes a combination of these problems is present, in which case a compound solution is needed; i.e., fix well and then use a non-acid second fix or wash well. It's really pretty simple...
I use a variety of papers, but usually a neutral-tone image on bright-white base (Seagull G when it was around, Emaks till I run out, Gallerie, Fomabrom, MC110, Fomabrom Variant, etc.). I have never had a problem with staining when transferring directly to the selenium toner from the second fix. As mentioned already, I use Ilford Hypam or Rapid Fix 1+9 and fix for 1.5-2 minutes per bath. Many have reported staining when using these products in the stronger 1+4 dilution. I suspect that the stronger dilution is more acid, which causes the staining. The weaker dilution seems to work just fine for me. Note however, that I have fixed already in fix 1 for 1.5-2 minutes and washed thoroughly, which removes all but traces of the unwanted silver compounds (since I always use fresh fix for bath one, I imagine that my prints are pretty close to archivally processed at this point, fix 2 being insurance and prep for the toning). I imagine this contributes to my never getting staining.
As for storing "half-fixed" prints... Yes, if the prints are really underfixed, they will darken upon exposure to light. However, the large majority of the fixing happens in the first bath (especially if you keep it fresh) so even after fix 1, the prints are fixed enough that light will not cause any printing out of the remaining thiosulfate complexes.
The non-image silver compounds in a print undergo a number of reactions when fixing, changing first from silver halides to insoluble silver thiosulfate compounds and then to soluble compounds that can be washed out (gross oversimplification, but that's all we need here). The first fix does the vast majority of this work if it's fresh (if not all), leaving only a few intermediate compounds to be fixed to completion so they can be washed out. That hardly exhausts the second fixer (which is why you can use it for fix 1 after its capacity has been reached). Point being: after fix 1, prints can be safely exposed to a bit of light with no ill effects as long as fix 1 wasn't completely exhausted. I've been doing this for years with no problems.
Back on topic: There are a number of workflows possible when toning. Doing everything in one session is the favorite of many. I would recommend:
Dev - Stop - Fix 1 (in whatever fixer you like) - 10-minute Rinse - Fix 2 (in a neutral-enough or alkaline fixer) - Selenium Toner - HCA - Wash, etc.
Of course, you could wash completely between fix 2 and toner and use a more acid second fix. The only downside is the extra time needed for the wash since you still have to wash again after toning.
Or, you can do as I do, and split your workflow into "printing session" and "toning session." I print for several days and collect the finished prints (they've all had fix 1 in fresh fix and a thorough wash). I then edit them down to just the "keepers" (I often make several variations - performances if you will - of the same print and live with them for a while till I've decided which I like best; the others are discarded. When I have enough for a batch (36 prints of one size), I'll do a toning session. Again, I use Rapid Fix 1+9 and transfer directly to the toner.
Replenishing and keeping selenium toner:
I have posted a lot on this, so I'll keep it short. Filter your working-solution toner before and after use with coffee filters or the like. Use it till toning times become inconveniently long and then replenish it with a few ounces of the stock (careful here, it's easy to overdo it. If you do, just dilute more). I have two gallons of toner (one "strong" and one "weak") that have been going for 10 years or more. They work just fine and my prints pass the residual silver and hypo tests every session.
AA recommended mixing selenium toner with wash aid (Hypo-Clearing Agent) to save a step. The problem here is that the wash aid exhausted well before the selenium toner, but the bath was discarded anyway. In those days, discarding a lot of expensive and dangerous selenium was common practice and good business for Kodak. It is much more economical and responsible to have a separate step for the wash aid.
Best,
Doremus