It is advised not to bring any metal in a 1 Tesla MRI scan room. But
what metal is attracted by the magnet, and what metal isn't ? (I'm
talking attraction, NOT image disturbance if the metal is in the
scanned area). Titanium ? Tungsten ? All kinds of alloys ? Silver ?
Gold ? Iron ? stainless steel ? Other types of stainless steel ? ...

iron nickle and cobalt are the ferromagnetic elements.  anything other
than materials containing those elements should be fine.  this
includes titanium, aluminum, copper, silver and on and on.  You should
be wary of most kinds of steel (it's made out of iron) but there are
non-magnetic types of stainless steel.  You can always check a
material by seeing if a refrigerator magnet will stick to it.

General rule of thumb:

300 (304 and 316 being most common) series stainless steels are
non-magnetic.  400 series stainless steels are magnetic.

18/10 (and similar) is also a common way of describing 300 series
stainless steels in cookware primarily.

Negative, racecar.  The magnet used for an MRI is significantly
stronger than that used for a refrigerator magnet.  Certain alloys may
have very small amounts of ferromagnetic elements.  So, it is possible
that a weak fridge magnet will not attract a given piece of metal, but
an incredible strong MRI magnet will...

Typically, though, the best rule of thumb is to just not take metal at
all into the room with you.

While the following story is extremely rare, here is what can happen
if there is magnetic metal in there...

http://www.mrireview.com/docs/mrideath.pdf

Negative, touf.

A magnetic field is a magnetic field.  There is only one kind.  I did
not say that the attraction of a refrigerator has to be strong.  But
if there is no attraction between a refrigerator magnet and a piece of
material, then it is safe to bring into the MRI room.  You may be
right that the best policy is simply not to bring in metal, or that it
is not a good idea to encourage the refrigerator magnet test because
people might not be careful enough to recognize a small attraction
force, but still, if the test is carried out properly, there is
nothing wrong with it.

...attraction of a refrigerator *magnet*...

You just said it yourself, racecar -- 

"not a good idea to encourage the refrigerator magnet test because
people might not be careful enough to recognize a small attraction
force"

The ultimate flaw in your test is that your measuring device (the
human) is probably not sensitive enough (not a matter of being
careful) to record any practical or feasible data.

Imagine the following experiment to put things into perspective:  you
want to find the weight of an Altoid mint.  You have three instruments
at your disposal-- 1) the load cell scales that go on giant cranes 2)
a bathroom scale, and 3) a pharmacy scale.

Which do you use?  Surely, placing an altoid on the first two scales
will produce some finite result, albeit, extremely small.  After all,
assuming they all work via load cells, even a one or two gram mass
does introduce a finite force, which will cause the voltage output to
change.  If they work via springs, then an object with a mass of a few
grams will introduce a small force that causes the springs to compress
ever so slightly, and causes the dial to move.  However, this test is
meaningless, because your scales are not sensitive enough for your
particular application.

Clearly, you want to use the pharmacy scale, as it has the best
sensitivity for your application.  You can be as careful as you want
when you put the Altoid on a bathroom scale, but you are still going
to read "0".

The only way around this is to get something like 10,000 Altoids, find
the cumulative weight, and divide by 10000 to find the average weight
of each one.  But isn't that effectively the same as getting a
stronger magnet?

All I'm saying is that there are oodles of alloys out there.  If a
particular metal is comprised of 1% iron, for instance, your fridge
magnet isn't going to do jack, because your hands will not detect any
notable attraction.  Get an MRI near it, and well, things are
different.

The original advice is correct: do not bring ANY metal into the MRI
room.  This is a safety issue.  Whether the metal is magnetic is
irrelevant.  Metals, whether magnetic or not, are conductors.  As
conductors, the MRI's external magnetic field will induce a
potentially large and dangerous eddy current in your hunk 'o metal. 
At this point, the eddy current sets up an opposing magnetic field
(which, numercally speaking, seeks to null the original applied
field).  These fields then interact and your metal will want to move,
perhaps violently.  Firthermore, depending on the resistance of the
metal, the induced eddy current will be disspated to some degree as
heat.  For a really cool demo of these effects, get yourself a
horseshoe magnet and an aluminum pie pan (with is non-magnetic, of
course).  Turn the pan upside down and balance it at its center with a
toothpick, so it floats in the air and rotates easily.  Hang the
magnet above the pan's center, and spin the magnet.  For best demo,
put a piece of clear plexiglass bewteen the magnet and the pie pan, to
eliminate any air drag between the two.  The pie pan will spin up and
follow the magnet's spin.  This is a good parlar trick and wins bets
at cocktail parties, too.

Here's another demo.  Get a hunk of copper tubing or pipe a foot or
two long.  Get one of those very strong rare earth magnets, preferably
cylindrical or spherical, that fits neatly and smoothly into the
tubing.  Hold the tube vertically and drop the magnet into the top. 
You'll be very surprised at how slow the magnet falls through the
tube!  Were it not for small parasitic loses in the copper due to its
resistance to the eddy current, the magnet would not fall at all, in
fact.