Is SHA-1 still practically secure under specific scenarios?
I would recommend phasing out SHA-1 in any scenario where collision-resistance of a hash is required, for there is a wide consensus that an attack with $2^{69}$ complexity would work, it would already be feasible by a resourceful entity, and attacks only get better.

I'm still confident that SHA-1 is preimage and second-preimage resistant for all practical purposes in the foreseeable future, when its full output is used. Nevertheless, I would prefer SHA-256 or RIPEMD-160 when they are possible options.

I'm still confident that HMAC with SHA-1 is secure in any scenario where HMAC's key is assumed secret, for all practical purposes in the foreseeable future; this is because an improved security argument for HMAC remains valid with weak assumptions on the underlying (round function of the) hash. I could recommend it in a MAC application where using a tarnished name is not an issue, and speed matters.



Update: A collision for SHA-1 was published in February 2017 (4 years after this answer was written). It took an effort comparable to $2^{63}$ hashes (less than the $2^{69}$ consensual by 2013, but more than the $2^{61}$ of the best claimed theoretical approach, which turned out to have a few rough edges). The phasing out of the first paragraph has become a must. The second and third paragraph of the answer remain valid.