The primary difference between "OC" SONET connectivity and Dedicated lines is that SONET provides a standby "protection" path. It is most useful for TDM connections that can't go down. For example, if you have more than 28 T1 trunks feeding a call center, then using an OC-3 as a transport is a viable method for handing off the trunks from a "reliable" infrastructure.
Furthermore, T1/DS1 and T3/DS3 dedicated circuits are focused more as point-to-point connections. SONET is a self-healing ring topology that prevents single-point issues if brought on premises properly. Thus SONET is a better choice for high capacity, high security, high performance network infrastructures.
Personally, I would say it isn't a specific "type" of application that's most important per se, so much as is the total "volume" of traffic. If, for example, you have a website that generates (or is expected to generate) 80-100Mbps of traffic, you'd probably want an OC3. If you have a sufficient number of users, even email or casual web browsing could eventually add up to the sufficient traffic to necessitate that large of a connection. But generally the volume.....especially for more "complex" activities (e.g. medical imagery, supply chain management networks, high security financial transaction storgae/networks, technical research and data manipulation, complex CAD projects, sophisticated high risk security systems)......is the more crucial part of the equation for deciding on a SONET solution.
Now if you flip the question.....just how important is data transmission, storage and back-up to your organization? How much data is involved, what is a geographically acceptable disaster recovery distance, and so on.....become the decision points. This makes more sense. The rest is simply algebra.
But I guess most companies only get to the algebra bit when it's too late. One could argue that all networked applications are appropriate for OC3, OC12, OC48 etc.....it's just a question of scale and security versus cost.
Consider future expansion needs as well when you make your decision. An OC3 may be your best bet given that as with other optical carrier levels (12, 48, etc), an OC-X (3 in this case) is burstable. In essence, you can start small in regards to your financial commitment of the bandwidth purchased and add capacity as your needs increase.
Given the above discussion....generally you'll most likely find high end network connections are for (examples):
- Server farms
- Video distribution (VoD, IPTV etc)
- Feeding super computers
- Interconnection with other networks
- Very high quality Imaging
- Big Business
- Banks
- Universities
- Hospitals
- Stock Exchanges
- Defense Networks
- Government
In short...the crux of the question is really that it's definitely a bandwidth (volume) issue over an application issue. You're talking huge amounts of bandwidth in use before you really need to move to an OC3+.
Now, for the vast majority of today's infrastructure.... much is still delivered over SONET/SDH infrastructure and Ethernet based services are still in their infancy. Once the far cheaper Ethernet based 100meg/GigE services are commonly made available.....the landscape may change. Yet again.