How (in)secure is a signature based on DSA with DSS1 (SHA1) in reality? As I found out the OpenSSL implementation provided by PHP (openssl_sign) only allows DSS1 as the hashing algorithm for DSA signatures (see also my related question). 

I’d prefer DSA over RSA in my use case because of the significantly shorter signatures. However, I have doubts regarding DSS1 which is basically SHA1 and therefore considered vulnerable.

The question is: Does this vulnerability also apply when used with DSA? After all the security of a signature is not (only) about finding a hash collision or fast brute-force hash calculations...

How secure or insecure would you say is using DSA with DSS1 when signing a license file for a software? Is it possible that someone can calculate the private key within a reasonable amount of time (having the public key, a message and its signature) or what’s a realistic risk?
I’m aware of the fact that calculations are getting faster and faster (GPUs, cloud based services, etc.) so please include your estimation for the next couple of years.
While no SHA1 collisions have been found, there are some attacks:


~$2^{60}$ collision attack. Estimated to cost around \$1-2 million currently in the cloud. Possibly economical soon, especially with specialized hardware.
Intractable preimage attacks like $2^{151}$ against reduced round variant, $2^{159}$ against full hash. (Cf. $2^{160}$ brute force on any 160-bit hash.)



  How secure or insecure would you say is using DSA with DSS1 when signing a license file for a software?


Like poncho wrote in a comment, using a collision attack against a signature requires the attacker having control over what is hashed. If you only sign messages you've created yourself, e.g. random data, there is no way to mount a collision attack.


  Is it possible that someone can calculate the private key within a reasonable amount of time (having the public key, a message and its signature) or what’s a realistic risk?


Even if a collision would be found, that would not compromise your key. A single collision would typically create a single valid-looking license file.

(A collision against a key signature, like against CAs, is a different matter because that allows one to create an arbitrary, valid, authorized key they can use to make their own signatures. That's why Microsoft is deprecating SHA-1 certificates.)


  I’m aware of the fact that calculations are getting faster and faster (GPUs, cloud based services, etc.) so please include your estimation for the next couple of years.


Current attacks could allow collision attacks in the next few years if you sign user controlled data.

Preimage attacks would require a breakthrough to become viable. Always a possibility, but there is still enough security margin that I wouldn't worry about it.

A note on preimage vs. second preimage:

Whether a preimage or a second preimage attack would be needed depends on your setup. If the scenario is that the attacker wants to create another signed license file knowing one, he would need a second preimage, which is possibly harder. If the scenario is that the attacker knows a signature and needs to create a license file to match it, any preimage will do.