Like many new hams, I have just one radio: an inexpensive 2m/70cm FM transceiver. It doesn't do SSB and it doesn't have a digital input. If I wire up a connection between the radio and my computer's sound card, are there any digital modes that I could reasonably take advantage of with this hardware?
I interpret your question to mean "what kind of cool digital stuff could I do with my inexpensive HT?" I can think of two digital activities that people commonly do with inexpensive handheld transceivers: APRS and Winlink 2000.
APRS is used to send and receive "tactical situation data". Most people use APRS to send GPS position reports from their cars, but it can also be used to send short person-to-person text messages, weather data, and all sorts of other things. (You can see what kind of information people are sending on APRS by pointing your browser at aprs.fi.) Winlink 2000 is a way to send and receive short emails over HF, VHF, or UHF: just the thing to help the larger community when disaster strikes. Both APRS and Winlink 2000 over VHF/UHF are based on AX.25 packet radio, which means that you would need some sort of Terminal Node Controller (TNC). The TNC is likely to cost more than your handheld, but as ham equipment goes they are considered relatively inexpensive. I use a Byonics Tiny Trak 4 for APRS; it's reasonably-priced and compact.
Check out APRS and/or Winlink 2000, and have fun!
We have been using traditional hf modes on a weekly 2 meter FM net for around 5 years. Participation varies from week to week. There is no weak signal advantage over a FM modulated signal. Our net is to encourage upgrading and to teach data modes. The local ARES group has recently started adding message and forms transfer using the fldigi suite to our ARES group to encourage wider spread adoption of these forms in our radio community for the same reason. SSTV, hell, PSK31, RTTY, and Olivia are among the most popular modes. We are now seeing activity on 10 meter data sub band locally. Give people a reason to upgrade and they will. Ham radio is more fun when it is more than just voice.
Nearly all common modes will work this way, however because of the way FM is generated and received, you get absolutely no advantage over voice, so it's only useful if you specifically want to transmit data for some reason.
Because of the much narrower envelope, a CW signal has between 9 and 12dB advantage over an SSB signal at the same power. Using PC audio to modulate an SSB signal as most digital modes do offers an additional 6dB more than CW, as they can be fantastically narrow. SSB is 2-4kHz wide on average, a clean CW signal should be less than 40Hz wide, and a properly generated JT9 signal may be less than 10Hz wide.
This means that a 100 watt CW signal will be about equivalent to a 1,000 watt SSB signal, and a JT9 signal will be equal to 2,000 Watts or more on SSB.
With FM, the envelope does not change width regardless of input, so all modes are equal.
If you can get audio between the radio and the computer, then there are a number of modes you can use.
Most common is AFSK, or audio frequency shift keying. This is some discrete number of audio tones, which are then FM modulated just as they would be if you whistled them into the microphone. The pitch of the tone encodes some digital bit or bits.
Usually, AFSK of a particular specification is used to encode the AX.25 protocol, which is then used for APRS and packet radio. Packet was common back in the day, but not so much today.
AFSK isn't an especially efficient modulation compared to designed-for-digital modulations like MSK. However, any FM radio can trivially be converted for AFSK operation (with an external computer), and you can find other hams using it. The same can not be said for most other modes.
It's also possible to take most digital modes meant to be used on HF SSB, and feed them into the audio input of an FM radio also. Apparently people do this with PSK31, although this is a horrible use of spectrum since FM-modulated-PSK31 spreads your transmitter power over a much wider bandwidth than PSK31 alone. There's also more noise power in this wider bandwidth, while the energy per bit transmitted is not increased. Consequently, the maximum error-free bitrate for PSK31-over-FM is much less than that of ordinary PSK31, or the theoretical maximum.
Well, there is a difference in regards to "mode" and "encoding" and "modulation"
You can "encode" a PSK "mode" on an FM "modulation", as long as "the other end" is using the same to demodulate and decode, you can make a contact.
Certain "modes" will work more efficiently with certain modulations, but it is nowhere written that you MUST use a particular modulation for any modes (with some restrictions and exceptions)
If you start using (the example given) PSK on top of an FM modulated signal, you might not get a lot of responses, as there not be a lot of people listening for that mode using that modulation. But let that not stop you ! Experimenting like this can be fun; but for first trials I would urge you to "line up the other end" to ensure you have someone listening, decoding and responding to your signal.