I'm just going to summarise here the points brought up and discussed in the thread that are currently tested and known for those who want to base their decisions on facts:
- Healer stat weights haven't been tested. Because of this we can't know the exact strenght of each secondary stat (Crit, Det and SpS) compared to each other.
- The formula for spellspeed is tested and known (source). It's calculated like this: 2.5 - (((Current Spell Speed - 334) / 26.5) * 0.01).
If you have 600 SpS (around the amount of SpS you'll have if you try to avoid the stat), your GCD is 0.1 seconds faster than the default 2.5: it's 2.4. If you have 1000 SpS your GCD is 0.25 seconds faster than the default: 2.25. So the difference between 1000 SpS and 600 SpS is 0.15 seconds (150 ms). This is the actual effect of the stat on your recast and cast times (the effect is same for your dodging movement).
- Comparing a healer with 600 SpS and 1000 SpS, the latter is able to squeeze in an extra cast only if they cast 15 spells in a row without any breaks or movement (16th cast would be the extra one, and it requires 36 seconds of straight casting without any breaks). Even in this situation SpS is only the stronger choise if that 16th cast from the SpS healer would be worth more than the bonuses the non-SpS healer would gain from Crit / Det on all the 15 casts combined (if the 16 casts from SpS healer would result in more healing / damage points than the 15 casts from Crit/Det healer). It's not currently known if that's the case.
- The effect of spellspeed on HoTs and DoTs has been tested by Dervy on WHM (source). 1 point of SpS boosts WHM HoTs / DoTs by around 0.000125, while 1 point of Det boosts WHM HoTs / DoTs by 0.000137. So we know Det is a bigger boost for HoTs and DoTs than SpS at least on WHM, but not by a lot. Crit is likely to boost both the most, but I haven't found a tested number to compare.
- For Piety (which of course doesn't boost your healing or DPS power directly) each point of Piety increases the maximum MP pool by ~6.5 for AST and the in-combat MP refresh rate is a base of 2% of your maximum MP per tic (source).
- For Accuracy, the current caster cap for A8S is around 600 (source).
Please post if you have found any more relevant numbers, formulas and comparisons that aren't included in here (or if you notice an error in my sources or calculations)!
Edit 6.10.2016: FFXIV 3.01 - 3.38 WHM/AST Theorycrafting (Av En of Midgardsormr) and discussion on Reddit