What are the advantages to a Class A amp & what are the trade offs?

Depending on your equiptment and you ears you will appreciate Class A more or less.  For me there is a relaxing and smoothness character to Class A.
I moved from a 540 watt Class D to 60 watt Class A to drive my Maggie 3.7i's.  I liked the Class D over the Class A/B Levinson. The 60 watt Class A has the current to drive the Maggies as loud as my wife will allow but not to the volume of the Class D.  I dont miss the Class D.  The Class A has much more umph and detail to the low end.  The Class A sounds better at all volumes.
I was worried about the heat but in my 20 x 25 room in Southern Ca. it has not been an issue.  I leave them on 24/7. The room is a little warm on warm days but it gets that way in the summer anyway.  I also have 2 tube preamps with a combined 10 tubes that are both on when I listen.  In a small room it may be an issue.
If you can afford Class A you wont likly go back to A/B or D.

@sfischer1 

Perhaps a good stereo in an office space, for break time, would revitalize the love of good music reproduction?

I have one, but it is only my office.  Actually I have an office cat and a Hi-Fi in my office.  Also a Barista coffee machine close at hand.
The Hi-Fi is single source - a Technics SL1210 turntable.  Other components are Ayon Pre, ATC power and Spendor D1 standmounts.  To be honest there is not great synergy but the sound is passable.  It sure looks cool though, and clients love the LPs dotting the office.  In order to up to cool factor I need to have some tubes showing.  The Ayon Pre is a tube based pre-amp but the tubes are all internal.  I want glowing bottles!

MPG on home audio?   Best music?

I listen to nothing but Class A amps now. I have three single ended troide amps using the 45 tube for 2 watts per channel powering a home made set of speakers using a 15" Altec woofer, a two cell Altec compression horn and a RAAL ribbon tweeter. The least efficient driver is the RAAL at 93db/watt/meter. The horn is 114 and the woofer is 102. On the other end of my room, sharing the source, is a pair of Sound Lab A-2 electrostatics. When I built the Altecs, I was using a Musical Fidelity Class AB amp, 300 watts into 8 ohms, 600 watts into 4, on the Sound Labs. I was very happy with that setup but the Altec speakers turned out so well I could no longer listen to the Sound Labs. I've owned the Sound Labs since 1985. I like them. I didn't want to abandon them.

In the early '80's I had the chance to hear the then new Mark Levinson system with the ML-2 amps. That session so impressed me that I remember it to this day. For the longest time I thought I could not use the ML-2s with my speakers since they were only 25 watts pure push-pull Class A. As it turns out, the ML-2 was designed on and for electrostatic speakers. I bought a pair and life is wonderful again. They power my speakers easily. They sound every bit as magical as I remembered from 38 years ago.

I have two other Class A amps; a British (now Musical) Fidelity A-1 and an Amp Camp Amp designed by Nelson Pass and built as a kit. Here is the main limitation, they are low power. They are inefficient. My ML-2s draw 400 watts of power to produce their 25 watts. They do not get particularly hot since they are half heat sink. The tube amps draw about 75 watts to output 2. They are no more of a space heater than any other tube amp. The ACA, I have never measured the current draw, make 6 single ended watts of transistor power, it gets warm but not too hot to touch. The real hot one is the A-1, people claim it is Class AB but it draws 80 watts from the wall at both idle and it's full power of 20 watts. Tim de Paravicini, the designer, says it is Class A, push pull. It has the sorriest heat sink, the top of the amp is ribbed and is the heat sink. It does get too hot to touch for very long.

In case you are worried about the power draw, I calculate my big amps cost one penny per hour to operate--for both of them. And in case you didn't catch my opinion, Class A is the only way to go. I can't guarantee all Class A amps are great, but it is a good place to start.

jspohn - I own an EAR 890 70w Class A zero feedback amp.  It does not have sufficient power for my Legacy Focus speakers which has a tough impedance and 6-12" woofers.  My highly modified Dynaco ST 70 with 30w Class A/B kills it on the focuses (tremendous bass control and dynamics).   When playing both amps on my Legacy Signature IIIs with an easier impedance and 6-10" woofers, the two amps sound very, very similar.  From this experience, I found that a Class A, very hot amp does not deliver into low impedance and/or big driver cones despite a smaller amps Class A/B design (not ultralinear but voltage regulated transformer tap design).  

I have several Yamaha CR620 35 watt Class A/B 1978 receivers that sound excellent and can drive the Focuses just fine (Tom Port of Better Records recommends this pairing).  I use them on smaller smaller speakers for my two video set-ups.