Can the data in a UDP packet be assumed to be correct at the application level?
UDP uses a 16-bit optional checksum. Packets which fail the checksum test are dropped.
Assuming a perfect checksum, then 1 out of 65536 corrupt packets will not be noticed. Lower layers may have checksums (or even stronger methods, like 802.11's forward error correction) as well. Assuming the lower layers pass a corrupt packet to IP every n packets (on average), and all the checksums are perfectly uncorrelated, then every 65536*n packets your application will see corruption.
Example: Assume the underlying layer also uses a 16-bit checksum, so one out of every 2^16 * 2^16 = 2^32 corrupt packets will pass through corrupted. If 1/100 packets are corrupted, then the app will see 1 corruption per 2^32*100 packets on average.
If we call that 1/(65536*n) number p, then you can calculate the chance of seeing no corruption at all as (1-p)^i where i is the number of packets sent. In the example, to get up to a 0.5% chance of seeing corruption, you need to send nearly 2.2 billion packets.
(Note: In the real world, the chance of corruption depends on both packet count and size. Also, none of these checksums are cryptographically secure, it is trivial for an attacker to corrupt a packet. The above is only for random corruptions.)
UDP uses a 16-bit checksum so you have a reasonable amount of assurance that the data has not been corrupted by the link layer. However, this is not an absolute guarantee. It is always good to validate any incoming data at the application layer, when possible.
Please note that the checksum is technically optional in IPv4. This should further drop your "absolute confidence" level for packets sent over the internet.
You are guaranteed only that the checksum is consistent with the header and data in the UDP packet. The odds of a checksum matching corrupted data or header are 1 in 2^16. Those are good odds for some applications, bad for others. If someone along the chain is dropping checksums, you're hosed, and have no way of even guessing whether any part of the packet is "correct". For that, you need TCP.