1. A computer-implemented method comprising: under control of a hardware computing device configured with specific computer executable instructions: obtaining, at a first domain name system (DNS) server from a client computing device, a first DNS query, wherein a first point of presence (POP) of a content delivery network (CDN) service provider includes the first DNS server, wherein the first DNS query is associated with a requested resource, wherein the requested resource is associated with a content provider, and wherein the CDN service provider is different from the content provider; determining, at the first DNS server, a routing mode from a plurality of available routing modes for providing the requested resource based at least in part on one or more criteria, wherein the plurality of available routing modes include two or more of a default routing mode, a sloppy routing mode, a regional anycast routing mode, or an anycast routing mode; dynamically selecting, at the first DNS server, (a) an alternative resource identifier associated with a second DNS server at a second POP of the CDN service provider or (b) an IP address of a cache component of the CDN service provider based at least in part on the determined routing mode, wherein the alternative resource identifier includes information for causing a second DNS query to resolve to the second DNS server of the CDN service provider, wherein the alternative resource identifier is a uniform resource locator (URL); and transmitting, to the client computing device, the alternative resource identifier or the IP address in accordance with the determined routing mode.