I'll chime in here as well. Although I don't want to muddy the waters, I do want to point out alternatives.
There are many, including myself, that find the Ilford sequence for optimum permanence inconvenient. It may be unreliable as well with papers other than Ilford's. My sequence is based on Kodak's recommendations and my personal testing.
The Ilford sequence relies on short fixing times in a strong fixer that is relatively fresh to reduce wash times. There are several downsides to this.
First, if you use just one fixing bath as outlined by Ilford, the capacity of the fixer is only 10 8x10-inch prints per liter of fixer before you have to mix new. I find this wasteful.
Second, if you use two-bath fixing and try to limit the time to 30 sec. in each bath in order to more efficiently use the fixer, it can be unwieldy, especially with larger prints. And, do you add the drain time to the fixing time? How long can the paper be exposed to fixer before too much soaks in to the paper base to require longer wash times? etc...
My tested and reliable sequence is as follows:
Develop, stop as usual.
Fix 1 in a rapid fixer (I use Hypam or Ilford Rapid Fix mostly, but others work just fine too) at "paper dilution," i.e., at the weaker dilution (1+9 for Ilford products) for 1.5-2 minutes
...
Fix 2 in the same as above.
Transfer the print directly to the selenium toner without an intermediate rinse. (I've never had staining when doing this except when the fixer was exhausted). Tone visually until the desired tone is reached, then...
Transfer the print directly to a wash-aid. I mix my own from sulfite and metabisulfite, but the Kodak HCA or Ilford products are good. I use a 10-minute treatment.
Transfer the print to the washer and wash for minimum 60 minutes in slowly running water.
Squeegee prints and place face-up on screens to dry.
Fixer capacity using this method is at least 36 8x10-inch prints per liter of fix 1. After that has been reached, discard fix 1 and replace it with fix 2. Mix a fresh second fixing bath.
... Often between fix one and fix two I'll wash and dry my prints so I can live with them for a day or three to cull the non-keepers. Then, after I've collected a batch (36 for me), I'll do a toning session starting with a 5-minute water soak and continuing with fix two and the rest.
I routinely check my prints for residual hypo and silver using the Kodak HT-2 and ST-1 tests respectively. My prints normally show zero discoloration with both these tests.
The problem with staining when transferring prints directly from fix to selenium toner seems to come from two causes: First, inadequate fixing before toning and second, transferring to the toner from a too-acid environment. Many conventional fixers are more acid than the Ilford products (possibly Bill's problem) and certainly more than the many alkaline fixers. The alternative to transferring directly from fixer to toner is to thoroughly wash prints before toning. A quick rinse (even 5-10 minutes) won't usually help.
FWIW, I've only had stains when transferring from Ilford fixers (1+9) to the selenium toner when I inadvertently way overused my second fix. Refixing in fresh fixer and then toning solved the problem.
As mentioned above, mixing selenium toner with wash-aid is an older practice when discarding selenium was not seen as irresponsible (or uneconomical). While you may need to do this with the Ilford sequence as Michael points out, IMO it's just another reason to not use the Ilford sequence. I re-use and replenish my selenium toner and only mix it with water. I have jugs of toner that have been going for 10 years or more and work just fine. Search here for my methods if you're interested.
Best,
Doremus