There are really three kinds of rigid foam panels you are going to have to choose from - Polyisocyanurate, Extruded Polystyrene (XPS), Expanded Polystyrene (EPS).
Before choosing, you should know exactly what you expect it to do, to make sure you walk away with the right one. They are all petroleum-based products but their characteristics, performance and ecological impacts vary significantly.
Rated at R6-6.5 per inch, but don't count on that. Most insulations actually perform a bit better the colder it gets but polyisocyanurate breaks that rule. As of about 15°C its performance starts to deteriorate, and badly. By the time you get down to the -20s Celsius it's nowhere near that. It can be a great product to use as long as you keep it warm, which is an odd thing to say about insulation.
The news of its R value petering out when you need it most still hasn't permeated entirely through the building industry, so you still see it being installed occasionally on the exterior of walls. It won't offer nearly the thermal protection you think it will in the dead of winter, and it may cause moisture damage due to its lack of permeability.
XPS - Extruded Polystyrene:
Rated at R5 per inch, but it will off-gas and lose a bit of performance over time. It will act as a vapour retarder (and become even less moisture permeable the thicker it is - 1 inch is about 1 perm, 2 inches about .5 perms); when taped it can act as an air barrier; it does not absorb moisture, nor is it affected adversely by it
Note: 1 perm and 60 ng are U.S. and Canadian equivalent rates of permeability, below that rate of permeability classifies a material as a type II vapour retarder, suitable for residential construction.
XPS works great in pretty much any circumstance above or below grade, wet or dry. Regrettably, the hydro fluorocarbons (HFCs) most commonly used as blowing agents are far more damaging to the climate than those used with other rigid foam boards. Some manufacturers speak of a coming transition to less harmful blowing agents; that will be great news when it happens. HFCs have a global warming potential (GWP) that is 1430 times worse than carbon.
EPS - Expanded Polystyrene:
Rated at R4 per inch; it's more permeable to air and moisture than XPS. Two inches of EPS has a moisture permeability rate of between 60 and 75 ng (1 to1.25 perms), which is on the cusp of qualifying it as a type II vapour retarder, but on the more 'breathable' side of the scale.
For reference sake, the traditional 6mil polyethylene vapour barrier has a permeability rating of 3.4 ng, making it about 18 times more vapour resistant than building codes allow.
The permeability of EPS can be handy at times if you want to add insulation to an existing wall assembly but are worried about trapping moisture, like retrofitting the exterior of buildings with additional insulation. Though to be absolutely sure you may be better with a mineral wool board which lets moisture pass right through.
The lower R value of EPS compared to XPS is in a way compensated for by having a higher R value per dollar, as it is somewhat cheaper. If you're not worried about losing an inch of space here or there, you'll get a higher R value with EPS for the same amount of money, albeit with a thicker wall.
The performance of EPS may drop slightly when it's wet (reports I've seen indicate somewhere in the area of 10-15 %, so nothing too catastrophic), it will also dry out just as quickly as it got wet and return to its original performance. But there is nothing wrong with putting a little effort into keeping it dry if you can. The GWP of expanded polystyrene blowing agents is about 7 times worse than carbon, but that's a lot less than being 1430 times worse like XPS is.
An additional feature with Polyisocyanurate:
Polyisocyanurate comes with a layer of foil on each side to keep the gases in, so there is the potential to solve a bit of a growing problem in wall assembly durability. Foil is a vapour barrier and a very good one at that, it fact it stops even more moisture than the normal 6 mil polyethylene normally used. So if you use it on the interior of a stud wall, you won't need to add an additional vapour barrier.
Here is the fun part - Since there is foil on either side of the panel, you end up with a harmless second vapour barrier, which is usually heresy in building design. But this can help in summer months when there is a risk of the vapour drive reversing due to air conditioning during hot humid weather. Any inward-bound moisture would be stopped at that inner layer of foil, which will be warmer than the foil on the other side, so you reduce your risk of summertime condensation.
That foil is the reason it can be problematic on the exterior, as you would be adding an exterior vapour barrier where you likely don't want one.
On the good news side, the GWP of blowing agents in Polyiso is similar to those in EPS, and in the right circumstances its R value is significantly higher, which deservedly or not helped earn it the reputation of being the 'greenest' foam. It can be a great choice when kept above freezing and away from moisture - so above grade for sure, and it makes a great interior thermal break when it's kept a bit warmer by batt insulation in stud cavities.
Being petroleum based should not result in foam being condemned by green builders on principle alone; it should be looked at in perspective. There are other great types of insulated sheathing (mineral wool and fiberglass to name two) and each will have their own benefits, drawbacks, carbon footprint and embodied energy through manufacturing, so even the greenest of the green will have some measurable impact. It takes energy to save energy, and manufacturing insulation is arguably one of the more noble things we currently do with fossil fuels.
Mineral wool / Rock wool
The only rigid board insulation that is not a petroleum-based foam product shown in the main photo above is the brown one, which is mineral wool. It is a recycled product made with the stone dust from industrial blast furnaces.
Like any particulate, it is best not to inhale the fibers while installing it, so like any batt insulation we recommend using eye protection and a mask, not to mention gloves to prevent itching. As an insulation it is non-toxic, and unlike foam products there are no blowing agents that will off-gas into your home. That means it can lead to cleaner indoor air quality, and the R value will never change, whereas foam products will deteriorate over time as the gases escape.
Mineral wool is rated at about R 4, it is unharmed by moisture, it is not a vapour barrier at any thickness, nor is it an air barrier. It also provides better soundproofing, and being made with stone, it is fireproof.
Not being affected by moisture makes it is a great choice for basements, either interior or exterior. It is also something of a fool proof exterior board insulation when renovating, as it will not trap moisture from the interior inside the wall.
In batt form is it a common alternative to fiberglass for its higher R value, and easier installation (at least for doing a good job) It does however cost more than fiberglass per batt.
Our personal preference due to its recycled content and versatile applications is mineral wool, but as for the foam products - polyisocyanurate gets top marks for being 'eco' if you can handle its moody disposition. EPS foam is versatile and in the middle ground for performance, financial and ecological cost, and XPS foam is a top performer but comes with some unfortunate baggage. As soon as XPS completes its transition to less harmful blowing agents, I'm sure it will be welcomed into the green building community.