Single transverse mode means that the beam is 'tem00' or a gaussian shape (more likely a gaussian stretched along on direction, this still counts as tem00). Most low powered (<100mw) laser diodes fall into this category.
Single longitudinal mode means that the laser outputs only one narrow range of wavelengths of light (contrary to popular belief, no laser outputs only a single frequency, they all emit a finite range of frequencies). Most lasers operate with several 'modes', so that if you were to look at the output on a sensitive enough spectrometer you would see several peaks separated by a fraction of a nanometer apart (ex, a typical 660nm diode laser will have 2-10 modes separated by about 0.05nm). Very few diode lasers fall into this category, unless they have been very carefully engineered as such, and even then they usually require precise temperature/current control to keep them operating in a single mode regime. Luckily, for the most part the only people that need this are holographers, it has no benefit for lasershow applications.
As far as what wavelengths are available in single transverse mode, you can get
* 375nm (extremely expensive, power up to 20mw) UV
* 405nm purple (very cheap, power up to 500mw)
* 450nm blue (moderately expensive, up to 50mw)
* 520nm green (moderately expensive, up to 70mw)
* 638nm red-orange (moderately expensive, up to
110mw)
* 660nm red (very cheap, up to 500mw
There are also a few specialty diodes available in 473nm-488nm which are MUCH more expensive than the 450nm ones if you really need the 'turquiose-ish' color, as well as a range of wavelengths in the red region (635-685nm) which vary from cheap to extremely expensive. And of course a large number of diodes available for IR wavelengths (785nm, 808nm, 830nm, 980nm, 1310nm, 1550nm, etc)
