With the demise of Advance, it seems appropriate to try to put together a complete list of what trains were produced. This is the latest version of my attempt. I'd be grateful for any corrections or additions. I've focused exclusively on the trains, whether they were issued individually or part of sets. I haven't tried to cover sets as such or any other things in the Advance range.

Product numbers are those that appear on the box. Where more than one produc[size=medium][/size]t has been issued with that number I've followed the BPT precedent and added the suffix N or NN ... to the later issues. This does not appear on the box.

Info in the asterisked items is what I've been able to confirm from my own collection. Main sources for the rest:
List provided by Nigels (thanks Nigels)
List on Hobby Search website (thanks Super for the reference)

Numbered issues
AS-01* Series 0 Shinkansen
AS-01N* N700A Shinkansen Nozomi
AS-02* Series 500 Shinkansen
AS-03* Type 923-3000 Dr. Yellow
AS-04* Series E231-500 Yamanote Line (Product title as on box, but also described as Bandai Yamanote)
AS-05* Series 485 Express
AS-05N* EF510 Red Thunder
AS-06* Nankai Rapi:t
AS-06N* Series E3 Shinkansen Toreiyu (Product title as on box, but also described as Toreiyu Tsubasa)
AS-07* Series 700 Shinkansen
AS-07N* Kintetsu Premium Express Shimakaze
AS-08* Meitetsu Panoramacar
AS-08N Steam Locomotive Type C57 No.1 "SL Yamaguchi"
AS-09* Series 700 Shinkansen Hikari Railstar
AS-09N* Limited Express Sleeping Cars "Cassiopeia" with EF510
AS-09NN* Keikyu 1000 (Also referred to as the aluminium train. The prototype is aluminium, not the model!)
AS-10* Series N700 Shinkansen Mizuho/Sakura
AS-11 Series E233 Keihin Tohoku Line
AS-12* Series 100 Shinkansen
AS-12N* Keisei Skyliner (Title as on box but also described as Keisei Skyliner Airport Express)
AS12NN* Hankyu Series 1000
AS-13* Series E3-0 Shinkansen (White/silver livery. Hobby Search describes as Shinkansen Komachi)
AS-14* Series 165 (Also described as Express Train Series 165)
AS-14N D51 200 No. 200 (Steam loco, blue carriages)
AS-15* Series E259 Narita Express
AS-16* Series E4 Shinkansen Max
AS-17* Series 200 Shinkansen
AS-17N* Series E235 Yamanote Line
AS-18 Series E233 Chuo Line (Q: Is this the same train as in the E231 and E233 double set?)
AS-18N* Series E233 Shonan Color

Unnumbered and special editions
Series E233 System Saikyo Line

500 Type EVA (Product linked to the Evangelion animé series)
Series E231-500 Yamanote line
Limited Express Sleeping Cars "Cassiopeia" with EF510 "Hokutosei" Color

Trains issued as part of sets
Series E5 Shinkansen Connect & double track point rail set (Hobby Search illustration shows E5 is AS-02N Hayabusa)
Series N700-3000 Shinkansen Entry Set

Series E231 Sobu Line & Series E233 Chuo Line double set
Series E2 Yamabiko/Series E3 Tsubasa connect and guide rail set
Series E6 Shinkansen Connect & Bridge Approach Rail set
Series E6 Shinkansen Komachi IR control set (Same train as E6 & bridge approach set)
W7 System Hokuriku Shinkansen "Kagayaki" & IR control set
Series L0 Superconducting Maglev and overpass rail set
Starter Set Shinkansen Series E5
Starter set Steam Locomotive D51 200 Entry Set (Blue carriages)